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hi = g ki SHU i = 1..m 2UD VLD s LO SL SLFFROR k i s

6L YXROH GLPRVWUDUH FKH H q XQ JUXSSR FLFOLFR JHQHUDWR GD g ,QIDWWL VH FRVu QRQ IRVVH HVLWHUHEEH XQ HOHPHQWR

hi = g ki ∈ H FRQ k i = qs + r H 0 ≤ r < s q LQWHUR 2VVLD HVLVWHUHEEH XQ HOHPHQWR LO FXL HVSRQHQWH QRQ q PXOWLSOR GL s '·DOWUD SDUWH SHU OD SURSULHWj GL FKLXVXUD GL H VL DYUHEEH FRVD DVVXUGD LQ TXDQWR r < s

hi g − qs = g qs + r g − qs = g r ∈ H

3DJ

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n

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$EHOLDQLWj GL XQ JUXSSR FLFOLFR

8Q JUXSSR FLFOLFR G q DEHOLDQR ,QIDWWL VH G q FLFOLFR HVVR ULVXOWD JHQHUDWR GD XQ VROR HOHPHQWR

g ∈ G SHUWDQWR VH g1 ∈ G H

g 2 ∈ G VHJXH FKH HVLVWRQR GXH LQWHUL (n, m) WDOL FKH

g1 = g n H g 2 = g m GD FXL

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•

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Hg = {hg , ∀h ∈ H , g ∈ G} = h1 g , h2 g ,....., hnh g ,O ODWHUDOH GHVWUR YHULILFD OH VWHVVH SURSULHWj GHO ODWHUDOH VLQLVWUR 8Q VRWWRJUXSSR N GL XQ JUXSSR G YLHQH GHWWR VRWWRJUXSSR QRUPDOH VH LO ODWHUDOH GHVWUR FRLQFLGH FRQ LO ODWHUDOH VLQLVWUR gN = Ng ∀g ∈ G 3HUWDQWR LQ FDVR GL JUXSSL QRUPDOL VL SDUOD VHPSOLFHPHQWH GL ODWHUDOL VHQ]D VSHFLILFDUH VH VLDQR GHVWUL R VLQLVWUL 3HU LQGLFDUH FKH N q XQ VRWWRJUXSSR QRUPDOH GL G VL XVD OD VHJXHQWH QRWD]LRQH N G PHQWUH SHU LQGLFDUH FKH G DPPHWWH N FRPH VRWWRJUXSSR QRUPDOH VL XVD OD QRWD]LRQH G N /D GHILQL]LRQH GL VRWWRJUXSSR QRUPDOH SXz HVVHUH SRVWD DQFKH QHOOD VHJXHQWH IRUPD ILVVDWR g ∈ G H ∀n ∈ N YDOH gng

−1

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n = n *

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hnh −1 ∈ N ⊂ H SRLFKp N q VRWWRJUXSSR GL G H YDOH OD SURSULHWj GL FKLXVXUD

GD FXL VHJXH

hnh −1 ∈ H N H

OHPPD

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GD FXL VHJXH

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( gNg 1 N )( g 2 N ) = gN ( g1 Ng 2 N )

• (VLVWHQ]D GHOO·HOHPHQWR LQYHUVR O·HOHPHQWR LQYHUVR GL gN q GDWR GD g

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g ∈ G LO JUXSSR FLFOLFR e = g 0 , g 1 , g 2 ,....g p −1 JHQHUDWR GD g q XQ VRWWRJUXSSR GL G FKH QRQ SRWHQGR HVVHUH XQ VRWWRJUXSSR SURSULR GHYH QHFHVVDULDPHQWH FRLQFLGHUH FRQ G VWHVVR LO TXDOH TXLQGL ULVXOWD HVVHUH XQ JUXSSR FLFOLFR GL RUGLQH p GRYH p q O·RUGLQH GHOO·HOHPHQWR

,QROWUH VH

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g '

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2PRPRUILVPL HG LVRPRUILVPL WUD JUXSSL 6XSSRQLDPR RUD FKH (G ,$) (G ′, # ) VLDQR GXH JUXSSL H O·DSSOLFD]LRQH Φ : G → G ′ XQ RPRPRUILVPR WUD HVVL 3HU LOOXVWUDWH OD SURSULHWj GL Φ GL PDQWHQHUH OD VWUXWWXUD DOJHEULFD 3DJ

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g ∈ G VLD XQ LQVLHPH FKH KD DQFRUD OD VWUXWWXUD GL JUXSSR ,QIDWWL GHWWL g , h, t WUH TXDOVLDVL HOHPHQWL GL G VL KD • Φ( g )# Φ(h) = Φ( g $ h) ∈ Im Φ ⊆ G ′ YDOH OD SURSULHWj GL FKLXVXUD •

[Φ( g )# Φ(h)]# Φ(t) = Φ( g $ h)# Φ(t) = Φ( g $ h $ t) =

= Φ[ g $ (h $ t)] = Φ( g )# Φ (h $ t) [Φ ( g )# Φ ( h) ]# Φ (t) = Φ ( g )# Φ (h $ t) = Φ ( g )# [Φ ( h)# Φ (t )]

YDOH OD SURSULHWj DVVRFLDWLYD •

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•

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1HO FDVR LQ FXL Φ IRVVH XQ LVRPRUILVPR VHJXH Im Φ ≡ G ′ LQ TXDQWR Φ q VXULHWWLYD H TXLQGL XQ

LVRPRUILVPR WUDVIRUPD WXWWR O·LQVLHPH G LQ G ′ PDQWHQHQGRQH LQROWUH OD VWUXWWXUD GL JUXSSR ,QILQH HVVHQGR XQ LVRPRUILVPR DQFKH LQLHWWLYR VL SXz GLUH FKH VH GXH JUXSSL VRQR LVRPRUIL UDSSUHVHQWDQR VRVWDQ]LDOPHQWH OR VWHVVR JUXSSR LQ FXL VHPSOLFHPHQWH QHO SDVVDJJLR GDOO·XQR '

DOO·DOWUR VL HVHJXH XQD VHPSOLFH WUDQVFRGLILFD GHO QRPH GL RJQL HOHPHQWR g ∈ G LQ Φ ( g ) ∈ G $G HVHPSLR VH XQ JUXSSR G q LVRPRUIR DG XQ JUXSSR DEHOLDQR K DQFKH G ULVXOWD DEHOLDQR ,QIDWWL GHWWL g1 H g 2 GXH HOHPHQWL GL G H k1 H k 2 GXH HOHPHQWL GL K VH Φ : K → G q O·LVRPRUILVPR VL KD g1 = Φ ( k1 ) g 2 = Φ ( k 2 ) GD FXL g1 g 2 = Φ ( k1 )Φ ( k 2 ) = Φ ( k1 k 2 ) = Φ ( k 2 k1 ) = g 2 g1

'DO SXQWR GL YLVWD QRWD]LRQDOH GXH JUXSSL G H G ′ LVRPRUIL VL LQGLFDQR FRQ OD VHJXHQWH QRWD]LRQH G ≈ G′

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Φ q OD WRWDOLWj GHJOL HOHPHQWL GHO JUXSSR G FKH WUDVIRUPDWL GD Φ PL SRUWDQR DOO·HOHPHQWR QHXWUR GHO JUXSSR (ImΦ, # ) 2UD DIILQFKp Φ VLD XQ LVRPRUILVPR HVVHQGR JLj XQ RPRPRUILVPR VXULHWWLYR SHU LSRWHVL RFFRUUH FKH VLD DQFKH LQLHWWLYR LQ PRGR GD YHULILFDUH OH FRQGL]LRQL GL ELLHWWLYLWj OD FRQGL]LRQH GL LQLHWWLYLWj VL LPSRQH SRQHQGR FKH KerΦ ≡ {eG }

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−1

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−1

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G

•

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= eG ' eG ∈ KerΦ

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Φ( ghg −1 ) = Φ ( g )Φ(h)Φ ( g −1 ) = Φ ( g )Φ (eG )Φ ( g −1 ) = Φ( g )Φ ( g −1 ) = Φ ( gg −1 ) = Φ (eG ) GD FLz VHJXH LQILQH

ghg −1 ∈ KerΦ

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KerΦ ≡ N LQ TXDQWR N q O·HOHPHQWR QHXWUR GL G / N SHUWDQWR RJQL VRWWRJUXSSR QRUPDOH SXz HVVHUH DVVRFLDWR DO QXFOHR GHOO·RPRPRUILVPR GHILQLWR GDOOD

/HJDPH WUD O·LPPDJLQH GL XQ RPRPRUILVPR H JUXSSL TXR]LHQWH

K H VLD Im(Φ) ≡ Φ(G) ⊂ K O·LPPDJLQH GL G GRYXWD D Φ 6L YXROH GLPRVWUDUH FKH Φ(G ) q LVRPRUIR D G / Ker (Φ)

6LD Φ : G → K XQ RPRPRUILVPR WUD LO JUXSSR G HG LO JUXSSR

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( g1 , g 2 ) ∈ G k1 k 2 = Φ ( g1 )Φ ( g 2 ) = Φ ( g1 g 2 ) ∈ Φ (G ) • (VLVWHQ]D GHOO·HOHPHQWR QHXWUR e K = Φ (eG ) ∈ Φ (G ) • (VLVWHQ]D

GHOO·HOHPHQWR

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FRQ

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Φ ( KerΦ ) ≡ e K SHU GHILQL]LRQH • JOL HOHPHQWL GL G SRVVRQR HVVHUH SDUWL]LRQDWL QHJOL m + 1 ODWHUDOL GHILQLWL GD KerΦ RVVLD VL KD

G = {eKerΦ , g1 KerΦ , g 2 KerΦ,....,.g m KerΦ}

•

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Im(Φ) ≡ Φ(G) q FRVWLWXLWD GDJOL m + 1 HOHPHQWL Φ ( g i )

Im(Φ ) ≡ Φ (G ) = {Φ ( g 0 ), Φ ( g1 ),...., Φ ( g m )}

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Φ : Φ (G ) → G / Ker (Φ ) Φ ( g i ) → g i KerΦ

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/·DSSOLFD]LRQH Φ q • ,QLHWWLYD LQ TXDQWR VH IRVVH g i KerΦ

Φ( g i ) ≠ Φ( g j ) SHU GHILQL]LRQH g i KerΦ ≠ g j KerΦ LQIDWWL VH

= g j KerΦ Φ( g i KerΦ) = Φ( g j KerΦ)

Φ( g i )Φ( KerΦ) = Φ( g j )Φ( KerΦ) GD FXL Φ( g i )e K = Φ( g j )e K Φ( g i ) = Φ( g j ) • 6XULHWWLYD LQ TXDQWR ∀ g i KerΦ H GHWHUPLQDWR GD Φ ( g i )

,QILQH Φ FRQVHUYD OH OHJJL GL FRPSRVL]LRQH LQ TXDQWR

Φ[Φ ( g i )]Φ Φ ( g j ) = ( g i KerΦ )( g j KerΦ ) = g i g j KerΦ = Φ Φ ( g i g j ) = Φ Φ ( g i )Φ ( g j )

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)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR > @ HN / N

≈ N /( N ∩ H )

GRYH HN = {hn : h ∈ H , n ∈ N } 3HU GLPRVWUDUH WDOH WHRUHPD VHJXLDPR L VHJXHQWL SDVVL SDVVR N q XQ VRWWRJUXSSR QRUPDOH GL HN 4XDQWR VRSUD DIIHUPDWR VHJXH GDO OHPPD VXL JUXSSL TXR]LHQWL 3HUWDQWR HVVHQGR N HN HVLVWH O·LQVLHPH TXR]LHQWH HN / N L FXL HOHPHQWL VRQR FRVWLWXLWL GDL ODWHUDOL hN FRQ h ∈ H SDVVR HN / N ≈ N /( N ∩ H ) 6L FRQVLGHUL O·DSSOLFD]LRQH Φ : H → HN / N N WDOH FKH Φ(h) = hN h ∈ H 7DOH DSSOLFD]LRQH WUDVIRUPD RJQL HOHPHQWR GL H QHO ODWHUDOH GHOO·LQVLHPH TXR]LHQWH HN / N HG q XQ RPRPRUILVPR VXULHWWLYR LQ TXDQWR • FRSUH WXWWL JOL HOHPHQWL HN / N • Φ(h)Φ(k ) = (hN )(kN ) = hkN = Φ(hk ) 4XDQWR SUHFHGH FL SHUPHWWH GL DIIHUPDUH DSSOLFDQGR LO WHRUHPD GLPRVWUDWR QHO SDUDJUDIR SUHFHGHQWH FKH YDOH OD VHJXHQWH UHOD]LRQH Φ( H ) ≈ H / KerΦ 0D HVVHQGR Φ VXULHWWLYD Φ( H )

≡ HN / N SHUWDQWR YDOH Φ( H ) = HN / N ≈ H / KerΦ

3HU FRPSOHWDUH OD GLPRVWUD]LRQH q DOORUD VXIILFLHQWH YDOXWDUH LO NHUQHO GL Φ $ WDOH VFRSR VL ULFRUGL FKH KerΦ = {h ∈ H : Φ ( h ) = N } HVVHQGR N O·HOHPHQWR QHXWUR GL HN / N

2UD VROR JOL HOHPHQWL n ∈ N VRQR WDOL FKH nN ∈ N RVVLD DSSDUWHQJRQR DO ODWHUDOH N 3HUWDQWR JOL HOHPHQWL h ∈ H H FKH IDQQR SDUWH GHO NHUQHO GL Φ GHYRQR DSSDUWHQHUH DQFKH D N RVVLD KerΦ ≡ H ∩ N H FLz FRQFOXGH OD GLPRVWUD]LRQH GHO WHRUHPD

6HFRQGR 7HRUHPD GHOO·LVRPRUILVPR

6LD G XQ JUXSSR FKH DPPHWWH GXH VRWWRJUXSSL QRUPDOL N HG H WDOH FKH H ⊂ N 9DOH OD VHJXHQWH UHOD]LRQH > @ G / N

≈

G/H N/H

1RWLDPR LQQDQ]L WXWWR FKH OD UHOD]LRQH q EHQ SRVWD LQ TXDQWR • HVVHQGR N HG H GXH VRWWRJUXSSL QRUPDOL GL G HVLVWRQR L GXH LQVLHPL TXR]LHQWL G / N H G / H • H N H TXLQGL HVLVWH DQFKH LO TXR]LHQWH N / H L FXL HOHPHQWL VRQR L ODWHUDOL GHOOD IRUPD nH FRQ n ∈ N 3HU YHULILFDUH TXHVWR IDWWR VL RVVHUYL FKH SUHVR n ∈ N HVLVWH XQ h ∈ H LO TXDOH DSSDUWLHQH DQFKH DG N HVVHQGR H ⊂ N WDOH FKH nhn

−1

∈ H SRLFKp H G /D VWUDWHJLD GHOOD GLPRVWUD]LRQH VHJXH XQD OLQHD DQDORJD D TXHOOD VHJXLWD QHO SUHFHGHQWH WHRUHPD FKH FRQVLVWH QHO WURYDUH XQ RPRPRUILVPR VXULHWWLYR Φ OD FXL LPPDJLQH Im Φ ≡ G / N HG LO FXL NHUQHO KerΦ ≡ N / H 6L ULFRUGL LQIDWWL FKH VH Φ : K → M q XQ RPRPRUILVPR VXULHWWLYR WUD L GXH JUXSSL K HG M YDOH OD VHJXHQWH UHOD]LRQH Φ( K ) ≈ K / KerΦ 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR 1HO QRVWUR FDVR GRYUHPPR GHILQLUH XQ RPRPRUILVPR Φ LQ FXL K ≡ G / H • Φ( K ) ≡ M = G / N • H SHU LO TXDOH YDOJD KerΦ ≡ N / H /·DSSOLFD]LRQH FKH YHULILFD WDOL SURSULHWj q GDWD GD Φ : G / H → G / N WDOH FKH ∀g ∈ G Φ( gH ) = gN ,QIDWWL Φ ULVXOWD HVVHUH XQ RPRPRUILVPR VXULHWWLYR LQ TXDQWR • Φ ( g1 H )Φ ( g 2 H ) = ( g1 N )( g 2 N ) = g1 g 2 N = Φ ( g1 g 2 H ) FRQ ( g1 , g 2 ) ∈ G •

DO YDULDUH GL g ∈ G OD Φ IRUQLVFH WXWWL ODWHUDOL GL G / N

/·HOHPHQWR QHXWUR GL G / N q LO ODWHUDOH N FKH SXz HVVHUH JHQHUDWR VROR GDL ODWHUDOL GL G / H GHOOD IRUPD nH FRQ n ∈ N 7DOL ODWHUDOL DOWUL QRQ VRQR FKH JOL HOHPHQWL GL N / H SHUWDQWR SRVVLDPR FRQFOXGHUH FKH KerΦ ≡ N / H FRPSOHWDQGR OD GLPRVWUD]LRQH

$OFXQH SURSULHWj HG DOFXQL OHPPL VXL JUXSSL QRUPDOL H TXR]LHQWL

$EHOLDQLWj GHO TXR]LHQWH GL XQ JUXSSR DEHOLDQR

6LD G XQ JUXSSR DEHOLDQR H N G DOORUD G / N q DEHOLDQR ,QIDWWL VL FRQVLGHULQR GXH HOHPHQWR ( g1 , g 2 ) ∈ G VL KD

( g1 N )( g 2 N ) = ( g1 g 2 ) N = ( g 2 g1 ) N = ( g 2 N )( g1 N ) G / N ULVXOWD DEHOLDQR

1RUPDOLWj GHOO·LQWHUVH]LRQH GL JUXSSL QRUPDOL

6LD G XQ JUXSSR H VLDQR N H K GXH VRWWRJUXSSL QRUPDOL GL G DOORUD H = N ∩ K q WDOH FKH H N H H K ,QIDWWL VDSSLDPR FKH H q VRWWRJUXSSR VLD GL N VLD GL K ,QROWUH GHWWL h ∈ H H TXLQGL h ∈ N H h ∈ K n ∈ N H k ∈ K VLD KD •

•

nhn −1 ∈ N SHU OD FKLXVXUD GL N FRPH VRWWRJUXSSR GL G nhn −1 ∈ K SHUFKp K G −1

,Q GXH SUHFHGHQWL SXQWL GLPRVWUDQR SHUWDQWR FKH nhn ∈ N ∩ K = H GD FXL VHJXH H N 6FDPELDQR GL UXROL GL K HG N VL GLPRVWUD OD VHFRQGD UHOD]LRQH GL QRUPDOLWj

1RUPDOLWj GHOO·LQWHUVH]LRQH WUD XQ VRWWRJUXSSR QRUPDOH H XQR QRQ QRUPDOH

6LD G XQ JUXSSR H VLDQR N H K GXH VRWWRJUXSSL G H VLD N G DOORUD H = N ∩ K q WDOH FKH H H K ,QIDWWL VDSSLDPR FKH H q VRWWRJUXSSR VLD GL N VLD GL K ,QROWUH GHWWL h ∈ H H TXLQGL h ∈ N H h ∈ K n ∈ N H k ∈ K VLD KD

khk −1 ∈ N SHUFKp N G SHU OD FKLXVXUD GL K FRPH VRWWRJUXSSR GL G −1 $OORUD khk ∈ H GD FXL H K • •

OHPPD

6LD N G H G / N LO JUXSSR TXR]LHQWH VLD LQROWUH K / N XQ VRWWRJUXSSR GL G / N 6L YXROH GLPRVWUDUH FKH K q XQ VRWWRJUXSSR GL G GRYH K = {g ∈: gN ∈ K / N } 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR 6L RVVHUYL SHU LQFLVR FKH N ⊂ K DOWULPHQWL QRQ DYUHEEH VHQVR FRQVLGHUDUH LO TXR]LHQWH K / N ,QIDWWL • SHU GHILQL]LRQH O·HOHPHQWR (e) XQ GL G GHYH DSSDUWHQHUH DQFKH D K LQ TXDQWR

eN = N ∈ K / N HVVHQGR K / N XQ VRWWRJUXSSR GL G / N FKH FRPH WDOH GHYH SRVVHGHUH O·HOHPHQWR QHXWUR N • •

gN ∈ K / N GD FXL g −1 N ∈ K / N g −1 ∈ K VH g 1 ∈ K H g 2 ∈ ( g1 N )( g 2 N ) = g 1 g 2 N ∈ K / N g 1 g 2 ∈ K

VH g ∈ K

6WUXWWXUD GHL VRWWRJUXSSL TXR]LHQWH

6LD H XQ VRWWRJUXSSR GL XQ JUXSSR G H VLD N ⊂ H XQ DOWUR VRWWRJUXSSR QRUPDOH GL G RVVLD YDOJD OD VHJXHQWH FDWHQD

N ⊂ ,

N G

⊂G

H ,

H sottogruppo di G

$EELDQR YLVWR FKH YDOH

N H

VL YHGD LO OHPPD GHO SDUDJUDIR ´6WUXWWXUD VRWWRJUXSSL QRUPDOLµ $OORUD KD VHQVR FRQVLGHUDUH L GXH JUXSSL TXR]LHQWL H / N H G / N H SHU HVVL YDOH FKH H / N q XQ VRWWRJUXSSR GL G / N ,QIDWWL GHWWL h H k GXH HOHPHQWL GL H VL KD FKH LQ H / N • (VLVWH O·HOHPHQWR QHXWUR N • 9DOH OD SURSULHWj GL FKLXVXUD LQ TXDQWR (hN )(kN ) = (hk ) N hk ∈ H SRLFKp H q VRWWRJUXSSR GL G (hk ) N ∈ H / N •

(VLVWH O·HOHPHQWR LQYHUVR GL hN ∈ H / N GDWR GD h

−1

N ∈ H / N

9LFHYHUVD FRQVLGHUDQGR TXDQWR GLPRVWUDWR QHO SUHFHGHQWH OHPPD VH H / N q XQ VRWWRJUXSSR GL G / N VHJXH H q XQ VRWWRJUXSSR GL G 3RVVLDPR TXLQGL FRQFOXGHUH VRWWR OH LSRWHVL

N ⊂ ,

N G

H ,

⊂ G RJQL TXR]LHQWH H / N q

H sottogruppo di G

VRWWRJUXSSR GL G / N 9LFHYHUVD RJQL VRWWRJUXSSR GL G / N q GHOOD IRUPD H / N

OHPPD

6LD N G H G / N LO JUXSSR TXR]LHQWH VLD LQROWUH K / N G / N $OORUD VL KD K G 6L RVVHUYL SHU LQFLVR FKH N ⊂ K N K DOWULPHQWL QRQ DYUHEEH VHQVR FRQVLGHUDUH LO TXR]LHQWH K / N ,QIDWWL K q XQ VRWWRJUXSSR GL G • • VLDQR gN ∈ G / N H kN ∈ K / N k ∈ K VHJXH

( gN )(kN )( g −1 N ) = gkg −1 N ∈ K / N gkg −1 ∈ K 9LFHYHUVD VH HVLVWH OD FDWHQD N K G VHJXH K / N G / N ,QIDWWL 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR • •

HVLVWRQR L GXH TXR]LHQWL K / N H G / N SUHVL GXH HOHPHQWL k ∈ K H g ∈ G VHJXH

( gN )(kN )( g −1 N ) = ( gkg −1 ) N ∈ K / N K / N G / N ∈K G

&ULWHULR GL PDVVLPDOLWj GHL VRWWRJUXSSL QRUPDOL

8WLOL]]DQGR L ULVXOWDWL GHO SDUDJUDIR SUHFHGHQWH SRVVLDPR ULFDYDUH XQ FULWHULR GL PDVVLPDOLWj GL XQ VRWWRJUXSSR QRUPDOH GL XQ JUXSSR GDWR 6LDQR N H G GXH JUXSSL WDOH FKH N G $OORUD N q PDVVLPDOH GL G VH H VROR G / N q VHPSOLFH 'LPRVWULDPR OD VXIILFLHQ]D OD SDUWH ´ VH µ LQ FXL G / N VHPSOLFH LPSOLFD N PDVVLPDOH 6LD G / N VHPSOLFH H VL VXSSRQJD SHU DVVXUGR FKH N QRQ VLD PDVVLPDOH $OORUD HVLVWH OD FDWHQD VHJXHQWH N H G 3HU TXDQWR RWWHQXWR QHO SDUDJUDIR SUHFHGHQWH FLz LPSOLFD O·HVLVWHQ]D GHO VRWWRJUXSSR SURSULR H / N GL G / N ,QROWUH HVVHQGR H G VHJXH GDO SUHFHGHQWH OHPPD FKH H / N G / N H FLRq G / N KD LO VRWWRJUXSSR QRUPDOH H G 7DOH FRQFOXVLRQH q DVVXUGD LQ TXDQWR SHU LSRWHVL G / N q VHPSOLFH HG DOORUD N GHYH HVVHUH PDVVLPDOH 'LPRVWULDPR OD QHFHVVLWj OD SDUWH ´VROR VH µ LQ FXL N PDVVLPDOH LPSOLFD G / N VHPSOLFH 6LD N PDVVLPDOH H VL VXSSRQJD SHU DVVXUGR FKH G / N QRQ VLD VHPSOLFH $OORUD G / N GHYH DPPHWWHUH XQ VRWWRJUXSSR QRUPDOH H / N H TXLQGL SHU LO SUHFHGHQWH OHPPD GXH GHYH YDOHUH H G '·DOWUD SDUWH N H LQ TXDQWR DEELDPR VXSSRVWR O·HVLVWHQ]D GL H / N GD FXL VHJXH O·HVLVWHQ]D GHOOD FDWHQD N H G FRVD DVVXUGD LQ TXDQWR N q PDVVLPDOH 3HUWDQWR SRVVLDPR FRQFOXGHUH FKH VH N q PDVVLPDOH G / N q VHPSOLFH

&RUROODULR DO FULWHULR GL PDVVLPDOLWj

6H G / N q XQ JUXSSR SULPR DOORUD N q PDVVLPDOH GL G ,QIDWWL VH G / N q SULPR QRQ SXz DYHUH VRWWRJUXSSL SURSUL H TXLQGL D PDJJLRU UDJLRQH QRQ SXz DYHUH VRWWRJUXSSL QRUPDOL SURSUL H SHUWDQWR ULVXOWD VHPSOLFH 1H FRQVHJXH GDOO·DSSOLFD]LRQH GHO FULWHULR SUHFHGHQWH OD PDVVLPDOLWj GL N

OHPPD

6LD G XQ JUXSSR H VLDQR N HG H GXH VRWWRJUXSSL GL G FRQ N G O LQVLHPH HN = {hn : h ∈ H , n ∈ N } q XQ VRWWRJUXSSR GL G ,QIDWWL HN YHULILFD OH WUH SURSULHWj GL XQ VRWWRJUXSSR • &KLXVXUD VLD ( x, y ) ∈ HN x = hu, y = kv FRQ (h, k ) ∈ H , (u, v) ∈ N N

∈ −1 xy = hukv = hk ( k uk , )v xy ∈ HN ∈H

∈N

•

(VLVWHQ]D GHOO·HOHPHQWR QHXWUR H HG N VRQR GXH VRWWRJUXSSL GL G FKH KDQQR LQ FRPXQH DOPHQR O·HOHPHQWR QHXWUR (e) SHU LO TXDOH YDOH e = e ⋅ e ∈ HN

•

(VLVWHQ]D GHOO·LQYHUVR VLD x ∈ HN x = hu FRQ h ∈ H , u ∈ N GD FXL VHJXH 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR −1 (hu −1h −1 ) ∈ HN x −1 = (hu ) −1 = u −1h −1 = h,

∈H

LQ TXDQWR

−1

h ∈ H,u

−1

∈N

∈ N H (hu −1h −1 ) ∈ N SHUFKp H HG N VRQR VRWWRJUXSSL H

N G

OHPPD

1HOOH VWHVVH LSRWHVL GHO OHPPD YDOH

N HN

,QIDWWL N YHULILFD OH WUH SURSULHWj GL VRWWRJUXSSR LQ TXDQWR SHU LSRWHVL q XQ VRWWRJUXSSR GL G HG LQROWUH N ⊂ HN LQ TXDQWR RJQL HOHPHQWR GL n ∈ N SXz HVVHUH SRVWR QHOOD IRUPD en ∈ HN HVVHQGR (e) O·HOHPHQWR QHXWUR 3HUWDQWR N q XQ VRWWRJUXSSR GL HN 6LD RUD x ∈ HN x = hu FRQ h ∈ H , u ∈ N HVLVWH XQ n ∈ N WDOH FKH

xnx −1 = h(unu −1 )h −1 ∈ N LQ TXDQWR SHU LSRWHVL N G

∈N

/D UHOD]LRQH VRSUD ULSRUWDWD SHUPHWWH SHUWDQWR GL FRQFOXGHUH FKH N HN H JDUDQWLVFH O·HVLVWHQ]D GHOO·LQVLHPH TXR]LHQWH HN / N L FXL HOHPHQWL VRQR FRVWLWXLWL GDL ODWHUDOL hN FRQ h ∈ H RVVLD HN / N q FRVWLWXLWR GDJOL VWHVVL ODWHUDOL FKH VL KDQQR FRQVLGHUDQGR VROR H LQYHFH GL HN

OHPPD 6LD G XQ JUXSSR H VLDQR H 1 H 2 N WUH VRWWRJUXSSL GL G WDOH FKH • • •

N G H 1 H 2 J 1 = H 1 ∩ N J 2 = H 2 ∩ N K1 = H 1 N / N K 2 = H 2 N / N

• $OORUD VL KD J 1 J 2

K1 K 2

HVLVWH XQ N H H 2 / H 1 WDOH FKH N H ≈ J 2 / J 1 H

H 2 / H1 ≈ K 2 / K1 NH

'LPRVWUD]LRQH SXQWR J 1 H J 2 FRPH LQWHUVH]LRQH GL VRWWRJUXSSL GL G ULVXOWDQR HVVHUH VRWWRJUXSSL GL G

H 1 ⊂ H 2 J 1 ⊂ J 2 3HUWDQWR J 1 ULVXOWD XQ VRWWRJUXSSR GL J 2 6L FRQVLGHULQR GXH HOHPHQWL j1 ∈ J 1 H j 2 ∈ J 2 VHJXH ,QROWUH HVVHQGR H 1 H 2

•

j 2 j1 j 2−1 ∈ N SHUFKp j1 ∈ N j 2 ∈ G H N G

•

j 2 j1 j 2−1 ∈ H 1 SHUFKp j1 ∈ H 1 j 2 ∈ H 2 H H 1 H 2

−1

$OORUD j 2 j1 j 2 ∈ H 1 ∩ N = J 1 J 1 J 2 'LPRVWUD]LRQH SXQWR ,O H OHPPD DVVLFXUDQR FKH OD GHILQL]LRQH GL K 1 H K 2 q EHQ SRVWD H ULVXOWD K 1 G / N

H 1 ⊂ H 2 H 1 N ⊂ H 2 N H 1 N / N ⊂ H 2 N / N 3HUWDQWR K 1 ULVXOWD XQ VRWWRJUXSSR GL K 2 ,QROWUH HVVHQGR H 1 H 2

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR 6L FRQVLGHULQR GXH HOHPHQWL k1 = h1 N ∈ K1 H k 2 = h2 N ∈ K 2 FRQ h1 ∈ H 1 H h2 ∈ H 2 VHJXH

k 2 k1k 2−1 = (h2 N )(h1 N )(h2 N ) −1 = (h2 h1h2−1 ) N ∈ H 1 N K 1 K 2

∈H1

'LPRVWUD]LRQH SXQWR 3DVVR GLPRVWULDPR OD VHJXHQWH LGHQWLWj H 2 ∩ H 1 N = H 1 ( H 2 ∩ N ) 3UHQGLDPR XQ TXDOVLDVL HOHPHQWR a ∈ H 2 ∩ H 1 N VHJXH

a ∈ H 2 H a ∈ H 1 N DOORUD SRVVLDPR SRUUH a = bc FRQ b ∈ H 1 H c ∈ N 2UD HVVHQGR H 1 H 2 RJQL HOHPHQWR GL H 1 DSSDUWLHQH DQFKH D H 2 SHUWDQWR b ∈ H 1 b ∈ H 2 ,QROWUH HVVHQGR H 2 XQ JUXSSR b ∈ H 2

b −1 ∈ H 2 H SHU OD SURSULHWj GL FKLXVXUD VHJXH

b −1a = b −1bc = c ∈ H 2 $OORUD

c ∈ N H FRQWHPSRUDQHDPHQWH c ∈ H 2 c ∈ H 2 ∩ N ,Q FRQFOXVLRQH a ∈ H 1 ( H 2 ∩ N ) HVVHQGR VFRPSRQLELOH QHO SURGRWWR bc FRQ b ∈ H 1 H c ∈ H 2 ∩ N SHUWDQWR VL KD H 2 ∩ H 1 N ⊂ H 1 ( H 2 ∩ N ) 9LFHYHUVD FRQVLGHULDPR XQ TXDOVLDVL HOHPHQWR a ∈ H 1 ( H 2 ∩ N ) VHJXH FKH SRVVLDPR SRUUH a = bc FRQ b ∈ H 1 H c ∈ H 2 ∩ N (VVHQGR H 1 H 2 VHJXH b ∈ H 1 b ∈ H 2 H SHUWDQWR SHU OD SURSULHWj GL FKLXVXUD VX H 2 VHJXH

a = bc ∈ H 2

,QROWUH SRLFKp b ∈ H 1 H c ∈ N VHJXH DQFKH

a = bc ∈ H 1 N 3HUWDQWR

a ∈ H 2 ∩ H1 N H1 (H 2 ∩ N ) ⊂ H 2 ∩ H1 N

,QILQH SRVVLDPR FRQFOXGHUH OD GLPRVWUD]LRQH GHO SDVVR QRWDQGR FKH H H 2 ∩ H 1 N ⊂ H 1 ( H 2 ∩ N ) H 1 ( H 2 ∩ N ) ⊂ H 2 ∩ H 1 N H 2 ∩ H 1 N = H 1 ( H 2 ∩ N ) 3DVVR 'HILQLDPR OD VHJXHQWH DSSOLFD]LRQH Φ : H2 → K2 FKH WUDVIRUPD ∀h ∈ H 2 QHO ODWHUDOH hN ∈ K 2 &RPH JLj RVVHUYDWR DO WHUPLQH GHO SUHFHGHQWH OHPPD HVVHQGR K 2 = H 2 N / N L VXRL ODWHUDOL VRQR JOL VWHVVL FKH VL RWWHQJRQR FRQVLGHUDQR L VROL HOHPHQWL GL H 2 LQYHFH GHJOL HOHPHQWL GL H 2 N

9HULILFKLDPR FKH O·DSSOLFD]LRQH Φ q XQ RPRPRUILVPR VXULHWWLYR ,QIDWWL OD VXULHWWLYLWj q LPSOLFLWD QHOOD GHILQL]LRQH SRLFKp DVVXPHQGR WXWWL JOL HOHPHQWL GL H 2 ULFRSUR WXWWL L ODWHUDOL GL K 2 PHQWUH SHU TXDQWR ULJXDUGD LO PDQWHQLPHQWR GHOOD VWUXWWXUD DOJHEULFD GHWWL (h, k ) GXH HOHPHQWL DSSDUWHQHQWL D H 2 VL KD

Φ(h)Φ(k ) = (hN )(kN ) = hkN = Φ(hk )

$YUHPR TXLQGL FKH Φ ( H 2 ) ≡ K 2

5LFRUGDQGR FKH K 1 ⊂ K 2 HVVHQGR K 1 K 2 VLDPR LQWHUHVVDWL D GHWHUPLQDUH O·LPPDJLQH LQYHUVD

Φ −1 ( K1 ) FKH UDSSUHVHQWD OD WRWDOLWj GHJOL HOHPHQWL h ∈ H 2 FKH IRUQLVFRQR L ODWHUDOL GL K 1 ⊂ K 2 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

Φ −1 ( K1 ) = {h ∈ H 2 : hN ∈ K1 ⊂ K 2 } 7DOL HOHPHQWL GHYRQR DSSDUWHQHUH VLFXUDPHQWH D H 2 H FRQWHPSRUDQHDPHQWH D H 1 N LQ TXDQWR DOWULPHQWL QRQ SRWUHEEHUR JHQHUDUH L ODWHUDOL GL K 1 VL RVVHUYL SHU LQFLVR FKH H 1 ⊂ H 1 N 3HUWDQWR VL KD

Φ −1 ( K1 ) ≡ H 2 ∩ H 1 N = H 1 ( H 2 ∩ N ) O·XOWLPR PHPEUR VL RWWLHQH DSSOLFDQGR LO SDVVR 3DVVR 'HILQLDPR OD VHJXHQWH DSSOLFD]LRQH Ψ : H 2 / H 1 → K 2 / K 1 7DOH DSSOLFD]LRQH WUDVIRUPD LO ODWHUDOL GL H 2 / H 1 FKH KDQQR OD IRUPD hH 1 FRQ h ∈ H 2 QHL ODWHUDOL GL K 2 / K 1 FKH KDQQR OD IRUPD kK 1 FRQ k ∈ K 2 LQ FXL ULFRUGDQGR OD GHILQL]LRQH

Φ GL FXL DO SDVVR H OD GHILQL]LRQH GL K 2 YLHQH SRVWR k = Φ(h) = hN 3HUWDQWR L ODWHUDOL GL K 2 / K 1 KDQQR OD IRUPD kK 1 = Φ ( h ) K1 2UD OD DSSOLFD]LRQH Ψ q VXULHWWLYD LQ TXDQWR Φ q VXULHWWLYD H TXLQGL DO YDULDUH GL h ∈ H 2 LQ WXWWL L PRGL VL RWWHQJRQR WXWWL L ODWHUDOL GL k ∈ K 2 H TXLQGL VL RWWHQJRQR DQFKH WXWWL L ODWHUDOL GL K 2 / K 1 ,QROWUH Ψ q XQ RPRPRUILVPR SRLFKp SUHVL GXH HOHPHQWL (h, k ) DSSDUWHQHQWL D H 2 VL KD GHOO·DSSOLFD]LRQH

Ψ ( hH 1 ) Ψ ( kH 1 ) = ( hN ) K 1 ( kN ) K 1 = Φ ( h) K 1Φ ( k ) K 1 = Φ ( h)Φ ( k ) K 1 = Φ ( hk ) K 1 = Ψ ( hkH 1 ) 'HWHUPLQLDPR RUD LO NHUQHO GL Ψ SHU GHILQL]LRQH HVVR q GDWR GDOOD WRWDOLWj GHL ODWHUDOL hH 1 ∈ H 2 / H 1 FKH IRUQLVFRQR O·HOHPHQWR QHXWUR GL K 2 / K 1 FKH q LQGLYLGXDWR GD K 1

Ψ q GHOOD IRUPD Φ ( h) K 1 H TXLQGL SHU RWWHQHUH K 1 QHFHVVDULDPHQWH Φ (h) GHYH HVVHUH XQ HOHPHQWR GL K 1 VWHVVR ,Q DOWUL WHUPLQL JOL HOHPHQWL hH 1 ∈ H 2 / H 1 GHO KerΨ VRQR WDOL FKH Φ ( h ) ∈ K 1 RVVLD VRQR OD 2UD K 1 FRPH FODVVH ODWHUDOH RWWHQXWD

−1

FRQWURLPPDJLQH Φ ( K1 ) ≡ $OORUD VL SXz DIIHUPDUH FKH

H 2 ∩ H 1 N = H 1 ( H 2 ∩ N )

{

}

KerΨ = hH 1 : h ∈ Φ −1 ( K1 ) 6L RVVHUYL LQILQH FKH HVVHQGR Φ OHPPD H SHUWDQWR H 1 LQVLHPH TXR]LHQWH

−1

( K1 ) ≡ H 1 ( H 2 ∩ N ) FL WURYLDPR QHOOH LSRWHVL GHO SUHFHGHQWH

−1

Φ ( K1 ) H L VXRL ODWHUDOL RVVLD LO KerΨ SRVVRQR HVVHUH HVSUHVVL FRPH KerΨ = Φ −1 ( K1 ) / H 1 = H 1 ( H 2 ∩ N ) / H 1

3DVVR 6LDPR DUULYDWL DOOD IDVH FRQFOXVLYD GHOOD GLPRVWUD]LRQH GHO SXQWR 9HULILFKLDPR LO JUXSSR N H q GDWR GDO NHUQHO GL Ψ

N H = KerΨ = Φ −1 ( K1 ) / H 1 ,QIDWWL • VDSSLDPR FKH OL NHUQHO GHJOL RPRPRUILVPL VRQR JUXSSL QRUPDOL GHO GRPLQLR SHUWDQWR VL KD Ψ : H 2 / H 1 → K 2 / K 1 KerΨ H 2 / H 1 •

−1

HVVHQGR KerΨ = Φ ( K1 ) / H 1 VXOO·LVRPRUILVPR VL KD

= H 1 ( H 2 ∩ N ) / H 1 DSSOLFDQGR LO SULPR WHRUHPD

KerΨ = Φ −1 ( K1 ) / H 1 = H 1 ( H 2 ∩ N ) / H 1 = ( H 2 ∩ N ) / H 1 ∩ ( H 2 ∩ N ) KerΨ = Φ −1 ( K1 ) / H 1 = ( H 2 ∩ N ) /( H 1 ∩ N ) = J 2 / J 1 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR GRYH HVVHQGR H 1 ⊂ H 2 H 1 ∩ ( H 2 ∩ N ) = ( H 1 ∩ N )

6FRPSRVL]LRQH GL XQ JUXSSR WUDPLWH VRWWRJUXSSL

6LD G XQ JUXSSR ILQLWR GL RUGLQH Ord (G )

= n g H VLDQR H H K GXH VRWWRJUXSSL GL G GL RUGLQH

ULVSHWWLYDPHQWH Ord ( H ) = n h H Ord ( K ) = nk 'HILQLDPR O·LQVLHPH SURGRWWR WUD L VRWWRJUXSSL H H K FRPH VHJXH HK = {hk : h ∈ H , k ∈ K }

3URSULHWj GHOO·LQVLHPH SURGRWWR

9DOJRQR OH VHJXHQWL SURSULHWj •

H m = H SHU m ≥ 1 FLz VHJXH GLUHWWDPHQWH GDO IDWWR FKH H q XQ VRWWRJUXSSR GL G H

YHUEDOH TXLQGL OD SURSULHWj GL FKLXVXUD

n

m

• VH H 1 q XQ VRWWRJUXSSR GL H VHJXH H 1 H = H SHU n ≥ 1 H m ≥ 1 DQFKH WDOL SURSULHWj VHJXRQR GLUHWWDPHQWH GDOOD SURSULHWj GL FKLXVXUD • VH h ∈ H DOORUD hH = H

7HRUHPD GL FRPPXWD]LRQH

'HWWR G XQ JUXSSR ILQLWR H GHWWL H H K GXH VXRL VRWWRJUXSSL HK q XQ VRWWRJUXSSR GL G VH H VROR VH HK = KH 'LPRVWULDPR OD VXIILFLHQ]D ´FRQGL]LRQH VHµ $ WDOH VFRSR VXSSRQLDPR FKH L VRWWRJUXSSL H H K FRPPXWLQR RVVLD FKH HK = KH ,Q WDOH FDVR YDOH OD SURSULHWj GL FKLXVXUD LQ TXDQWR GHWWL h1 H h2 GXH HOHPHQWL GL H k1 H k 2 GXH HOHPHQWL GL K VL KD

h1 k1 h2 k 2 ∈ ( HK )( HK )

VH HK

= KH VHJXH ( HK )( HK ) = ( KH )( HK ) GD FXL h1k1h2 k 2 = k1 h, 1 h2 k 2 = h k 1k 2 = hk ∈ HK , h∈H

k∈K

,QROWUH HVLVWH O·HOHPHQWR QHXWUR e ∈ HK LQ TXDQWR H H K HVVHQGR VRWWRJUXSSL GL G FRQWHQJRQR HQWUDPEL O·HOHPHQWR QHXWUR GL G (G LQILQH HVLVWH O·HOHPHQWR LQYHUVR GL RJQL HOHPHQWR GL HK ,QIDWWL VH h1 k1 ∈ HK VHJXH

(h1k1 )(h1k1 ) −1 = h1 k1k1−1 h1−1 = h1h1−1 = e e

H SHUWDQWR O·LQYHUVR GL ( h1 k1 ) q GDWR GD

(h1k1 ) −1 LQ TXDQWR (h1k1 ) −1 = k1−1h1−1 ∈ HK FRQ

h1−1 ∈ H H k1−1 ∈ K 5LVXOWD SHUWDQWR FRQFOXVD OD GLPRVWUD]LRQH GHOOD FRQGL]LRQH VXIILFLHQWH 'LPRVWULDPR OD QHFHVVLWD ´FRQGL]LRQH VROR VHµ 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR $ WDOH VFRSR VXSSRQLDPR FKH HK VLD XQ VRWWRJUXSSR GL G &RQVLGHULDPR L VHJXHQWL GXH HOHPHQWL h ∈ H H k ∈ K VLFFRPH HK q XQ VRWWRJUXSSR GL G

(hk ) −1 ∈ HK k −1 h −1 ∈ HK

H

h −1 ∈ H k −1 ∈ K k −1h −1 ∈ KH $OORUD DEELDPR GLPRVWUDWR FKH ILVVDWR FRPXQTXH XQ HOHPHQWR GL GHOO·HOHPHQWR LQYHUVR SXz VHPSUH HVVHUH PHVVR QHOOD IRUPD (hk )

KH H TXLQGL

−1

HK FKH SHU O·HVLVWHQ]D

WDOH HOHPHQWR DSSDUWLHQH D

HK ⊂ KH

9LFHYHUVD LQ PRGDOLWj DQDORJD D TXDQWR YLVWR VRSUD ILVVDWR XQ TXDOVLDVL HOHPHQWR (kh) VHJXH (kh)

−1

(VVHQGR HK

−1

∈ KH

= h −1k −1 ∈ HK H SHUWDQWR

KH ⊂ HK ⊂ KH H KH ⊂ HK VL SXz FRQFOXGHUH FKH HK = KH

5LVXOWD SHUWDQWR FRQFOXVD OD GLPRVWUD]LRQH GHOOD FRQGL]LRQH QHFHVVDULD

7HRUHPD GHO SURGRWWR

'HWWR •

G XQ JUXSSR ILQLWR GL RUGLQH Ord (G ) = n g

• H H K GXH VRWWRJUXSSL GL G GL RUGLQH ULVSHWWLYDPHQWH Ord ( H ) = n h H Ord ( K ) = nk •

T = H ∩ K LO VRWWRJUXSSR GL G RWWHQXWR GDOO·LQWHUVH]LRQH WUD H H K GL RUGLQH Ord (T ) = nt

,O QXPHUR GHJOL HOHPHQWL GHO SURGRWWR JUXSSL VRSUD GHILQLWL

HK q GDWR GD 9DOH OD VHJXHQWH UHOD]LRQH WUD JOL RUGLQL GHL

nh nk Ord ( H )Ord ( K ) RVVLD nt Ord (T ) 3URFHGLDPR DOOD GLPRVWUD]LRQH T ULVXOWD HVVHUH XQ VRWWRJUXSSR GL H SHUWDQWR H SXz HVVHUH VFRPSRVWR WUDPLWH L ODWHUDOL VLQLVWUL GL T H = {h1T , h2T ,....hl T } RSSXUH XWLOL]]DQGR DOWUD QRWD]LRQH l

H = ∪ hi T i =i

5LFRUGDQGR FKH L ODWHUDOL KDQQR OR VWHVVR QXPHUR GL HOHPHQWL GL T VL KD Ord ( H ) = lOrd (T ) RVVLD nh = nt l GD FXL

l=

nh nt

2UD VL KD

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR l

l

i =i

i =i

HK = ∪ hi TK = ∪ hi K LQ TXDQWR HVVHQGR T ⊂ K TK = K SHU OH SURSULHWj GHOO·LQVLHPH SURGRWWR 6L RVVHUYL LQROWUH FKH • RJQXQR GHJOL l LQVLHPL hi K KD n k = Ord (T ) HOHPHQWL • RJQL FRSSLD GL LQVLHPL hi K H h j K FRQ i ≠ j q GLVJLXQWD LQIDWWL VH IRVVH hi k s −1

VHJXH FKH h j

= h j kr

hi = k s k r−1 ∈ T GD FXL hi T = h j T FRVD DVVXUGD SHU OH SURSULHWj GHJOL LQVLHPL

ODWHUDOL 3RVVLDPR DOORUD FRQFOXGHUH HYLGHQ]LDQGR FKH LO QXPHUR GHJOL HOHPHQWL GHO SURGRWWR HK q GDWR GD

nk l =

nh nk nt

/HPPD

6LD G XQ JUXSSR ILQLWR H VLD H XQ VXR VRWWRJUXSSR 9DOH TXDQWR VHJXH GRYH K = gHg 'LPRVWUD]LRQH

−1

∀g ∈ G K = gHg −1 ≈ H

{

}

= ghg −1 : h ∈ H SHU g ∈ G ILVVDWR

K = gHg −1 ULVXOWD XQ VRWWRJUXSSR GL G LQ TXDQWR • FRQWLHQH O·HOHPHQWR QHXWUR SRLFKp e ∈ H HG e

= geg −1

• RJQL HOHPHQWR GL K KD XQ HOHPHQWR LQYHUVR LQ TXDQWR GHWWR k HOHPHQWR GL K VL KD ( ghg

−1

= ghg −1 FRQ h ∈ H XQ

)( gh −1 g −1 ) = e GD FXL gh −1 g −1 = k −1 ∈ K

• YDOH OD SURSULHWj GL FKLXVXUD SRLFKp k1

= gh1 g −1 ∈ K H k 2 = gh2 g −1 ∈ K FRQ h1 ∈ H H

h2 ∈ H VHJXH k1k 2 = gh1 g −1 gh2 g −1 = gh1h2 g −1 ∈ K 'HILQLDPR RUD OD VHJXHQWH DSSOLFD]LRQH ELXQLYRFD

Φ : H → gHg −1 −1

−1

FKH DVVRFLD DG RJQL h ∈ H XQ k = ghg = Φ (h) ∈ gHg 3HU GLPRVWUDUH FKH OD FRUULVSRQGHQ]D q ELXQLYRFD EDVWD RVVHUYDUH FKH −1

−1

• VH Φ ( h1 ) = Φ ( h2 ) VHJXH gh1 g = gh2 g GD FXL h1 = h2 SHUWDQWR O·DSSOLFD]LRQH q LQLHWWLYD • O·DSSOLFD]LRQH q LQROWUH VXULHWWLYD SHU LO PRGR LQ FXL q VWDWD GHILQLWD 9HULILFKLDPR RUD FKH OD Φ LQGLYLGXD XQ RPRPRUILVPR ,QIDWWL

Φ(h1h2 ) = gh1h2 g −1 = ( gh1 g −1 )( gh2 g −1 ) = Φ(h1 )Φ(h2 ) 3HUWDQWR HVVHQGR O·DSSOLFD]LRQH Φ XQ RPRPRUILVPR ELLHWWLYR ULVXOWD HVVHUH XQ LVRPRUILVPR H −1 TXLQGL YDOH gHg ≈ H SHU TXDOVLDVL g ∈ G

6LD •

7HRUHPD GL VFRPSRVL]LRQH GL )UREHQLXV

G XQ JUXSSR ILQLWR GL RUGLQH Ord (G ) = n g H g i ∈ G XQ HOHPHQWR GHO JUXSSR

• H H K GXH VRWWRJUXSSL GL G GL RUGLQH ULVSHWWLYDPHQWH Ord ( H )

3DJ

= n h H Ord ( K ) = nk

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

•

Ti = g i−1 Hg i ∩ K LO VRWWRJUXSSR GL G RWWHQXWR GDOO·LQWHUVH]LRQH WUD H H K GL RUGLQH Ord (Ti ) = ni

6LD Fi = Hg i K XQ LQVLHPH FRVu GHILQLWR

Fi = Hg i K = {g ∈ G : hg i k , h ∈ H , k ∈ K } 9DOH DOORUD OD VHJXHQWH VFRPSRVL]LRQH GHO JUXSSR G l

G = ∪ Hg i K i=i

HG LQROWUH l

nh n k i =1 ni

ng = ¦

'LPRVWUD]LRQH &RQVLGHULDPR GXH LQVLHPL Hg1K H HgK FRQ g ∈ G VH Hg 1 K = HgK VHJXH ∀h ∈ H , ∀k ∈ K hg1 k = hgk g1 = g SHU OD SURSULHWj GL VHPSOLILFD]LRQH $OORUD L GXH LQVLHPL Hg1K H HgK VRQR GLVJLXQWL 3HUWDQWR SRVVLDPR SURFHGHUH DG XQD VFRPSRVL]LRQH GHJOL HOHPHQWL GHO JUXSSR LQ LQVLHPL GHO WLSR Hg1K ,QIDWWL FRQVLGHULDPR XQ HOHPHQWR g1 ∈ G FKH GHWHUPLQD Hg1K VH Hg1K KD HVDXULWR WXWWL JOL HOHPHQWL GL G OD VFRPSRVL]LRQH q WHUPLQDWD DOWULPHQWL HVLVWH XQ HOHPHQWR g 2 ∈ G WDOH FKH g 2 ∉ Hg 1 K H SRVVLDPR GHILQLUH X DOWUR LQVLHPH Hg 2 K H FRVu YLD ILQR DO ULFRSULPHQWR GL WXWWL JOL HOHPHQWL GL G 5LVXOWD FRVu GLPRVWUDWR FKH l

G = ∪ Hg i K i =i

9DOXWLDPR RUD LO QXPHUR GL HOHPHQWL GHOO·LQVLHPH Fi = Hg i K s

Fi = Hg i K DYUj s HOHPHQWL GD FXL Fi = Hg i K = ∪ f i j j =i

s

2UD VL FRQVLGHUL

g i−1 Hg i K = ∪ g i−1 f i j WDOH LQVLHPH KD HVDWWDPHQWH s HOHPHQWL GLVWLQWL LQ TXDQWR j =i

VH IRVVH

g i−1 f i j

=

g i−1 f i r

j

r

VHJXH f i = f i

−1

−1

2UD g i Hg i K = ( g i Hg i ) K SXz HVVHUH SHQVDWR FRPH LO SURGRWWR GL GXH LQVLHPL • O·LQVLHPH K −1

• O·LQVLHPH g i Hg i FKH SHU TXDQWR GLPRVWUDWR QHO SUHFHGHQWH OHPPD ULVXOWD LVRPRUIR D H H TXLQGL KD nh HOHPHQWL −1

3HU LO WHRUHPD GHO SURGRWWR LO QXPHUR GL HOHPHQWL GL g i Hg i K q GDWR DOORUD GD ( ULFRUGDQGR FKH •

Fi = Hg i K H g i−1 Hg i K KDQQR OR VWHVR QXPHUR GL HOHPHQWL 3DJ

nh nk ni

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR l

•

G = ∪ Hg i K i =i

l

VL SXz FRQFOXGHUH FKH n g

nh n k i =1 ni

=¦

5HOD]LRQH GL FRQLXJLR WUD JOL HOHPHQWL GL XQ JUXSSR 6LD G XQ JUXSSR GXH HOHPHQWL (h, k ) ∈ G VL GLFRQR FRQLXJDWL VH HVLVWH DOPHQR XQ HOHPHQWR g ∈ G WDOH FKH

k = ghg −1 ⇔ kg = hg ,Q VRVWDQ]D GXH HOHPHQWL VRQR FRQLXJDWL VH HVLVWH XQ HOHPHQWR GHO JUXSSR FKH JOL SHUPHWWH GL SHUPXWDUH 6L RVVHUYL FKH k ROWUH DG HVVHUH FRQLXJDWR FRQ h SRWUHEEH DQFKH HVVHUH FRQLXJDWR FRQ XQ DOWUR

g~ ∈ G WDOH FKH k = g~ l~ g −1 ~ L TXDOL LQ JHQHUDOH /D GHILQL]LRQH GL FRQLXJLR QRQ LPSRQH O·XJXDJOLDQ]D GHJOL HOHPHQWL g H g

HOHPHQWR l ∈ G ,Q TXHVWR FDVR VL DYUHEEH O·HVLVWHQ]D GL XQ HOHPHQWR VDUDQQR GLYHUVL WUD ORUR

,O FRQLXJLR FRPH UHOD]LRQH GL HTXLYDOHQ]D 'LPRVWULDPR FKH LO FRQLXJLR q XQD UHOD]LRQH GL HTXLYDOHQ]D YHULILFDQGR OH WUH SURSULHWj VLPPHWULFD ULIOHVVLYD H WUDQVLWLYD ,QIDWWL VLDQR (h, k , l ) ∈ G YDOH •

3URSULHWj ULIOHVVLYD RJQL HOHPHQWR h q FRQLXJDWR FRQ VH VWHVVR ,Q TXHVWR FDVR O·HOHPHQWR g ≡ e HOHPHQWR QHXWUR GL G HVVHQGR he = eh

•

3URSULHWj VLPPHWULFD VH h q FRQLXJDWR FRQ k WUDPLWH O·HOHPHQWR g ∈ G DOORUD DQFKH k q FRQLXJDWR FRQ h WUDPLWH O·HOHPHQWR

•

g −1 ∈ G ,QIDWWL SHU LSRWHVL HVLVWH g ∈ G WDOH FKH

k = ghg −1 h = g −1kg 3URSULHWj WUDQVLWLYD VH h q FRQLXJDWR FRQ k H k q FRQLXJDWR FRQ l DOORUD h q FRQLXJDWR −1 ~kg~ −1 VHJXH l = ( g~g )h( g −1 g~ −1 ) = ( g~g )h( g~g ) −1 FRQ l ,QIDWWL k = ghg H l = g

/H FODVVL GL HTXLYDOHQ]D LQGRWWH GDOOD UHOD]LRQH GL FRQLXJLR GHWHUPLQDQR XQD SDUWL]LRQH GHJOL HOHPHQWL GL XQ JUXSSR FKH SRVVRQR TXLQGL HVVHUH VXGGLYLVL LQ FODVVL GL FRQLXJLR /D FODVVH GL HTXLYDOHQ]D FRUULVSRQGHQWH DOO·HOHPHQWR QHXWUR FRQWLHQH VROR WDOH HOHPHQWR −1

1HO FDVR GHL JUXSSL DEHOLDQL VH h q FRQLXJDWR FRQ k VHJXH k = ghg k = gg 3HUWDQWR RJQL FODVVH GL FRQLXJLR GL XQ JUXSSR DEHOLDQR FRQWLHQH XQ VROR HOHPHQWR

−1

h = eh = h

,O FRQFHWWR GL FHQWUDOL]]DQWH H GL FHQWUR 6L GHILQLVFH FHQWUDOL]]DQWH GL XQ HOHPHQWR k ∈ G O·LQVLHPH GHILQLWR FRPH VHJXH

{

}

C (k ) = g ∈ G : k = gkg −1 ,O FHQWUDOL]]DQWH C (k ) q XQ VRWWRJUXSSR GL G LQ TXDQWR •

&KLXVXUD VLDQR ( g 1 , g 2 ) ∈ C (k )

−1 −1 g1 g 2 k ( g1 g 2 ) −1 = g1 ( g 2 kg 2 ) g1 =∈ C (k )

k

•

(VLVWHQ]D GHOO·HOHPHQWR QHXWUR k = eke

•

(VLVWHQ]D GHOO·LQYHUVR g ∈ C (k )

−1

= k e ∈ C (k )

k = gkg −1 k = g −1kg g −1 ∈ C (k )

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR 6L GHILQLVFH FHQWUR GL XQ JUXSSR G O·LQVLHPH GHILQLWR FRPH VHJXH

Z (G ) = {g ∈ G : gk = kg , ∀k ∈ G} ,O FHQWUR Z (G ) q XQ VRWWRJUXSSR DEHOLDQR QRUPDOH GL G.

3URFHGLDPR FRPH DO VROLWR D YHULILFDUH OH SURSULHWj GL VRWWRJUXSSR • &KLXVXUD VLDQR ( g 1 , g 2 ) ∈ Z (G ) −1 −1 ∀k = G g1 g 2 k ( g1 g 2 ) −1 = g1 ( g 2 kg 2 ) g1 =∈ Z (G )

k

•

(VLVWHQ]D GHOO·HOHPHQWR QHXWUR ∀k = G k = eke

•

(VLVWHQ]D GHOO·LQYHUVR g ∈ Z (G )

−1

= k e ∈ Z (G)

∀k = G k = gkg −1 k = g −1kg g −1 ∈ Z (G )

,O IDWWR FKH Z (G ) VHJXH GDOOD GHILQL]LRQH VWHVVD GL FHQWUR SRLFKp L VXRL HOHPHQWL FRPPXWDQR FRQ WXWWL JOL HOHPHQWL GL G H TXLQGL FRPPXWDQR DQFKH WUD ORUR 9HULILFKLDPR RUD FKH Z (G ) G RVVLD VH g ∈ G HVLVWH h ∈ Z (G ) WDOH FKH ghg SHU GHILQL]LRQH h FRQ RJQL HOHPHQWR GL G GD FXL VHJXH gh =

hg = h = ghg

−1

−1

∈ Z (G ) ,QIDWWL

∈ Z (G )

/HJDPH WUD FHQWUDOL]]DQWH H FHQWUR

,O FHQWUDOL]]DQWH C (k ) GHILQLVFH O·LQVLHPH GHJOL HOHPHQWL GL G FKH FRPPXWDQR FRQ k PHQWUH LO FHQWUR

Z (G) GHILQLVFH O·LQVLHPH GHJOL HOHPHQWL GL G FKH FRPPXWDQR FRQ RJQL DOWUR HOHPHQWR GL

G

3HUWDQWR VH k ∈ Z (G ) HVVR FRPPXWD FRQ WXWWL JOL HOHPHQWL YLFHYHUVD

g ∈ G GD FXL VHJXH FKH C (k ) ≡ G H

k ∈ Z (G ) ⇔ C (k ) ≡ G

1HO FDVR GHL JUXSSL DEHOLDQL •

k = gkg −1 gk = gk ∀g ∈ G C (k ) = G ∀k ∈ G LQIDWWL QHO FDVR GHL JUXSSL

•

DEHOLDQR WXWWL JOL HOHPHQWL GL JUXSSR VRQR FRPPXWDWLYL Z (G) ≡ G

/·HTXD]LRQH GL FODVVH GL XQ JUXSSR ILQLWR

,QGLFKLDPR FRQ n( ki ) LO QXPHUR GL HOHPHQWL GHOOD FODVVH GL FRQLXJLR UDSSUHVHQWDWD GD k i ∈ G 6LFFRPH JOL HOHPHQWL GL XQ JUXSSR VL SRVVRQR SDUWL]LRQDUH LQ FODVVL GL FRQLXJLR DYUHPR GHWWR r LO QXPHUR GL WDOL FODVVL VL KD r

Ord (G ) = ¦ n(k i ) i =1

Z (G) q XQ VRWWRJUXSSR DEHOLDQR GL G OH VXH FODVVL VRQR FRVWLWXLWH GD XQ VROR HOHPHQWR RVVLD RJQL HOHPHQWR GL Z (G) q FDUDWWHUL]]DWR GDOO·DSSDUWHQHUH DG XQD FODVVH GL FRQLXJLR

2VVHUYDQGR FKH

FRVWLWXLWD VROR GD VH VWHVVR $OORUD VL SXz SRUUH r

Ord (G ) = Ord [Z (G )] + ¦ n(k i ) i =1

GRYH TXHVWD YROWD OD VRPPDWRULD GHYH LQWHQGHUVL DSSOLFDWD DOOH FODVVL UHODWLYH DJOL HOHPHQWL GHOO·LQVLHPH G − Z (G )

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

1XPHUR GL HOHPHQWL GL XQD FODVVH GL FRQLXJLR

&HUFKLDPR RUD GL GHWHUPLQDUH LO QXPHUR n(k ) GHJOL HOHPHQWL GHOOD FODVVH GL FRQLXJLR GL k ∈ G LQ IXQ]LRQH GHJOL RUGLQL GHO JUXSSR G H GHO FHQWUDOL]]DQWH C (k ) $ WDOH VFRSR VL SRQJD n = Ord (G ) G = {g1 , g 2 ,...., g n } • • •

m = Ord[C (k )] C (k ) = {g1,k , g 2,k ,...., g m,k }

LS C ( k ) = {g1C (k ), g 2 C (k ),...., g s C (k )} LQVLHPH GHL ODWHUDOL VLQLVWUL GHO FHQWUDOL]]DQWH C (k ) LQ FXL s LQGLFD LO QXPHUR GL ODWHUDOL

6L ULFRUGD FKH JOL HOHPHQWL GL XQ JUXSSR SRVVRQR HVVHUH SDUWL]LRQDWL QHL GLYHUVL ODWHUDOL H TXLQGL RJQL HOHPHQWR GL G DSSDUWLHQH DG XQ VROR ODWHUDOH g i C (k ) 3RLFKp Ord [C (k )] = m VHJXH

n = ms s =

n Ord (G ) = m Ord [C (k )]

,Q DOWUL WHUPLQL LO QXPHUR GL ODWHUDOL VLQLVWUL q GDWR GDOO·LQGLFH GHO FHQWUDOL]]DQWH C (k ) QHO JUXSSR

G 6L GHILQLVFDQR RUD OH VHJXHQWL TXDQWLWj

{

}

g i C (k ) g i−1 = ( g i g1,k )k ( g i g1,k ) −1 , ( g i g 2,k )k ( g i g 2,k ) −1 ,...., ( g i g m,k )k ( g i g m,k ) −1 SHU i = 1..n −1 9DOXWLDPR LO JHQHULFR HOHPHQWR GL g i C ( k ) g i ( g i g j ,k )k ( g i g j ,k ) −1 = g i ( g j ,k kg −j ,1k ) g i−1 = g i kg i−1 SHU j = 1..m

>g

−1 j , k kg j ,k

= k SRLFKp g

k −1 j , k ∈ C ( k ) H g i kg i q LO FRQLXJDWR GHOO·HOHPHQWR k

∈ G FRQ g i ∈ G @

−1

$OORUD SRVVLDPR FRQFOXGHUH FKH g i C ( k ) g i FRLQFLGH FRQ XQ VROR HOHPHQWR H SUHFLVDPHQWH FRQ LO −1

FRQLXJDWR g i kg i

{

}

g i C (k ) g i−1 = g i kg i−1 1RWLDPR RUD FKH VH g l ∈ g i C (k ) VHJXH g l

= g i g l ,k FRQ l ∈ [1, m] GD FXL

g l kg l−1 = g i g l ,k k ( g i g l ,k ) −1 = g i ( g l ,k kg l−,k1 ) g i−1 = g i kg i−1

k

RVVLD L FRQLXJDWL GL HOHPHQWL DSSDUWHQHQWL DG XQR VWHVVR ODWHUDOH FRLQFLGRQR 6L RVVHUYL TXLQGL FKH • RJQL ODWHUDOH VLQLVWUR SXz HVVHUH PHVVR LQ FRUULVSRQGHQ]D ELXQLYRFD FRQ XQ HOHPHQWR

g i C (k ) g i−1 g i C (k ) ↔ g i C (k ) g i−1 i = 1..s •

−1

RJQL LQVLHPH g i C ( k ) g i q LQ FRUULVSRQGHQ]D ELXQLYRFD FRQ XQ VROR HOHPHQWR FRQLXJDWR FRQ k LQ TXDQWR FRLQFLGH FRQ WDOH HOHPHQWR

3HUWDQWR VL SXz FRQFOXGHUH FKH LQ QXPHUR GL HOHPHQWL GHOOD FODVVH GHJOL HOHPHQWL FRQLXJDWL FRQ k q SDUL DO QXPHUR GHL ODWHUDOL VLQLVWUL GHO FHQWUDOL]]DQWH C (k ) FKH DEELDPR YLVWR q GDWR GDOO·LQGLFH GHO FHQWUDOL]]DQWH C (k ) QHO JUXSSR G 3DJ

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s=

n Ord (G ) = n( k ) = m Ord [C (k )]

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r

Ord (G ) i =1 Ord [C ( k i )]

Ord (G ) = ¦ n(k i ) Ord (G ) = ¦ i =1

'D FXL VHJXH

r

1 i =1 Ord [C ( k i )]

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{

}

N ( H ) = g ∈ G : gHg −1 = H

−1

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g ∈ N (H )

g −1 ∈ N ( H )

•

VH

•

g −1 gHg −1 g = g −1 Hg = H VH g1 ∈ N ( H )

VHJXH

LQ

TXDQWR

GD

g 2 ∈ N ( H )

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VHJXH VHJXH

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=H

YDOH OD SURSULHWj GL FKLXVXUD

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/HJDPH GL QRUPDOLWj WUD XQ JUXSSR HG LO VXR QRUPDOL]]DQWH

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−1

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S N (S )

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cH =

Ord (G ) Ord [ N ( H )]

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2VVHUYD]LRQL

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2UD VH

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7HRUHPL VXL VRWWRJUXSSL GL XQ JUXSSR ILQLWR 5LFRUGLDPR LO WHRUHPD GL /DJUDQJH FKH VWDELOH FKH O·RUGLQH GL XQ VRWWRJUXSSR GL XQ JUXSSR ILQLWR GLYLGH O·RUGLQH GHO JUXSSR VWHVVR ,Q JHQHUDOH QRQ q YHUR LO YLFHYHUVD RVVLD QRQ q GHWWR FKH XQ VRWWRLQVLHPH GL XQ JUXSSR FKH FRQWLHQH XQ QXPHUR GL HOHPHQWL FKH GLYHGH O·RUGLQH GHO JUXSSR VLD DQFKH XQ VRWWRJUXSSR GL WDOH JUXSSR (VLVWRQR SHUz GHL WHRUHPL FKH VWDELOLVFR GHL OHJDPL WUD FRQ OD 3DJ

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Ord (G ) = n = p1k1 p 2k2 .... plkl GRYH •

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•

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p k GHYH GLYLGHUH Ord (G)

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p2

p 2 FRQ p QXPHUR SULPR DOORUD G ULVXOWD DEHOLDQR

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p RSSXUH p 2

p 2 Z (G ) ≡ G

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r

p2 = p + ¦ 3URFHGHQGR VL KD

p=

1 1−α

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S VRWWRJUXSSL GL 6\ORZ k

k

k

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3DJ

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VRWWRJUXSSR LQ FXL O·HVSRQHQWH k i q LO SL JUDQGH HVSRQHQWH SRVVLELOH SHU FXL p i i GLYLGH Ord (G ) k +1

k

LQ DOWUL WHUPLQL p i i GLYLGH Ord (G ) PHQWUH pi i

QRQ GLYLGH Ord (G )

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7HRUHPD GL &DXFK\ 6LD G XQ JUXSSR ILQLWR GL RUGLQH Ord (G )

= n H VLD p XQ QXPHUR SULPR FKH GLYLGH Ord (G)

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&DVR DEHOLDQR 6LD GXQTXH G XQ JUXSSR DEHOLDQR WDOH

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LQ H H SRLFKp H q VRWWRJUXSSR GL G YDOH DQFKH LQ G 3DVVR 6XSSRQLDPR DOORUD FKH QHVVXQ VRWWRJUXSSR GL G DEELD O·RUGLQH GLYLVLELOH SHU

p

6L FRQVLGHUL DOORUD XQ VRWWRJUXSSR H GL G FKH QRQ VLD FRQWHQXWR LQ QHVVXQ DOWUR VRWWRJUXSSR SURSULR GL G GL RUGLQH PDJJLRUH 7DOH VRWWRJUXSSR FHUWDPHQWH HVLVWH LQ TXDQWR LO QXPHUR GL VRWWRJUXSSL GL G q ILQLWR H WUD WXWWL LQ VRWWRJUXSSL QH HVLVWH DOPHQR XQR FKH KD LO PDJJLRU QXPHUR GL HOHPHQWL ULVSHWWR DJOL DOWUL VRWWRJUXSSL RG DO SL SRWUDQQR HVVHUFL SL VRWWRJUXSSL FRQ OR VWHVVR QXPHUR PDVVLPR GL HOHPHQWL 6L FRQVLGHUL RUD XQ HOHPHQWR g ∉ H GHO JUXSSR G (VVHQGR G QRQ SULPR O·HOHPHQWR g JHQHUD

≠ H LQ TXDQWR SHU g ∉ H 2UD O·LQVLHPH HK ULVXOWD FHUWDPHQWH XQ VRWWRJUXSSR GL G LQ TXDQWR GHWWL k1 H k 2 GXH HOHPHQWL GL K H h1 H h2 GXH HOHPHQWL GL H XQ VRWWRJUXSSR K FLFOLFR GL G WDOH FKH K

• •

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−1

= h1k1h1−1k1−1 = h1h1−1 k1k1−1 = e e

•

e

YDOH OD FKLXVXUD ( h1 k1 )(h2 k 21 ) = ( h1 h2 )(k1 k 2 )

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H FKH QRQ SXz HVVHUH

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Ord (G ) / Ord ( H ) = Ord ( K ) / Ord [ H ∩ K )] 2UD VLFFRPH

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LQ H H SRLFKp H q VRWWRJUXSSR GL G YDOH DQFKH LQ G 3DVVR 6XSSRQLDPR DOORUD FKH QHVVXQ VRWWRJUXSSR GL G DEELD O·RUGLQH GLYLVLELOH SHU 5LFRUGLDPR O·HTXD]LRQH GHOOH FODVVL

p

r

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SHU OH LSRWHVL SRVWH

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p GLYLGH ¦ 2VVLD

p GLYLGH Ord[ Z (G)] 3DJ

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3ULPR 7HRUHPD GL 6\ORZ 6LD G XQ JUXSSR ILQLWR GL RUGLQH Ord (G )

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•

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k

p FKH GLYLGH Ord (G) H p

k +1

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FKH QRQ GLYLGH Ord (G ) k

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p k FKH GLYLGH Ord (G) H FRQ p k +1 FKH QRQ

GLYLGH Ord (G ) '

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h

p FRQ h ≤ k

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p h+1 p h LQ G '

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H GL G WDOH FKH p k GLYLGH Ord (H ) H TXLQGL

p k +1 QRQ GLYLGH Ord (H )

3DJ

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p k GHO JUXSSR H H WDOH JUXSSR K q DQFKH VRWWRJUXSSR

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p &RPH QHO FDVR GHOOD GLPRVWUD]LRQH GHO WHRUHPD GL &DXFK\ VL FRQVLGHUL O·HTXD]LRQH GHOOH FODVVL FRQ OR VWHVVR VLJQLILFDWR GHL VLPEROL LOOXVWUDWL LQ TXHOOD GLPRVWUD]LRQH r

Ord (G ) i =1 Ord [C ( k i )]

Ord (G ) = Ord [ Z (G )] + ¦

GRYH OD VRPPDWRULD GHYH LQWHQGHUVL DSSOLFDWD DOOH FODVVL UHODWLYH DJOL HOHPHQWL GHOO·LQVLHPH G − Z (G ) 2UD ULFRUGDQGR FKH • L FHQWUDOL]]DQWL C ( ki ) SHU i = 1..r VRQR WXWWL VRWWRJUXSSL GL G •

SHU OH LSRWHVL SRVWH

p k QRQ GLYLGH Ord [C (ki )] SHU i = 1..r PHQWUH GLYLGH Ord (G)

DOORUD VHJXH FKH O·RUGLQH GL RJQL FHQWUDOL]]DQWH C ( ki ) SXz DYHUH DO SL XQ IDWWRUH O·RUGLQH GL G KD XQ IDWWRUH

p k −1 PHQWUH

p k H SHUWDQWR

r Ord (G ) 1­ Ord (G ) ½ 1 ®Ord (G ) − ¦ ¾ = Ord [ Z (G )] = m LQWHUR p¯ p i =1 Ord [C ( ki )] i =1 Ord [C (k i )] ¿ r

p GLYLGH ¦ 2VVLD

p GLYLGH Ord[ Z (G)] H G DSSOLFDQGR LO WHRUHPD GL &DXFK\ VHJXH FKH Z (G) GHYH FRQWHQHUH XQ HOHPHQWR g ∈ G GL RUGLQH p FKH JHQHUD XQ VRWWRJUXSSR N GL Z (G ) GL RUGLQH p 2UD VLFFRPH Z (G ) q VRWWRJUXSSR GL G YDOH DQFKH • •

N VRWWRJUXSSR GL G N G LQ EDVH DOOD GHILQL]LRQH GL FHQWUR Z (G)

3RVVLDPR TXLQGL FRQVLGHUDUH LO JUXSSR TXR]LHQWH G / N LO FXL RUGLQH ULVXOWD GLYLVLELOH SHU HVVHQGR • •

p k −1

Ord (G) GLYLVLELOH SHU p k D VHJXLWR GHOO·LSRWHVL LQGXWWLYD Ord (G / N ) = Ord (G ) / Ord ( N ) = Ord (G ) / p

$OORUD VHPSUH SHU O·LSRWHVL LQGXWWLYD SRVVLDPR GHGXUUH FKH G / N DPPHWWH O·HVLVWHQ]D GL XQ VRWWRJUXSSR GL 6\ORZ S GL RUGLQH HG DVVXPHUj XQD IRUPD GHO WLSR

p k −1 ,O VRWWRUJUXSSR S q FRVWLWXLWR GD p k −1 ODWHUDOL GL G / N

{

}

S = eN , g1 N , g 2 N ,......g p k −1 −1 N 2UD VH LQGLFKLDPR FRQ S O·LQVLHPH

{

}

S = g ∈ G : gN ∈ S DSSOLFDQGR OH SURSULHWj HG L OHPPL VXL JUXSSL QRUPDOL H TXR]LHQWL SUHFHGHQWHPHQWH GLPRVWUDWL VL GHGXFH FKH S q XQ VRWWRJUXSSR GL G 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR 3HU TXDQWR ULJXDUGD O·RUGLQH GL S VL RVVHUYL FKH WDOH VRWWRJUXSSR q FRVWLWXLWR GDOO·XQLRQH GHJOL HOHPHQWR FRQWHQXWL QHL ODWHUDOH GL S RVVLD

g ∈ G ⇔ g ∈ eN ∪ g1 N ∪ g 2 N ∪ ...... ∪ g p k −1 −1 N

2UD LO QXPHUR GL ODWHUDOL q SDUL D

p k −1 HG RJQL ODWHUDOH FRQWLHQH p HOHPHQWL SHUWDQWR Ord ( S ) = pp k −1 = p k

H VL SXz FRQFOXGHUH OD GLPRVWUD]LRQH RVVHUYDQGR FKH S LQGLYLGXD LO S VRWWRJUXSSR GL 6\ORZ GL G

7HRUHPD GHL S VRWWRJUXSSL

p m XQ S VRWWRJUXSSR GL 6\ORZ GL XQ JUXSSR ILQLWR G GL RUGLQH Ord (G) = n FRQ p QXPHUR SULPR FKH GLYLGH Ord (G ) 6LD S GL RUGLQH

l

$OORUD G FRQWLHQH S VRWWRJUXSSL GL RUGLQH p FRQ l < m LQWHUR /D GLPRVWUD]LRQH GL TXHVWR WHRUHPD q GHO WXWWR DQDORJD D TXHOOD GHO SULPR WHRUHPD GL 6\ORZ LQ FXL LQYHFH GL EDVDUFL VXOO·LSRWHVL LQGXWWLYD GHOO·HVLVWHQ]D GL XQ VRWWRJUXSSR GL 6\ORZ GL RUGLQH

p k WDOH

p k GLYLGH O·RUGLQH GL G H p k +1 QRQ GLYLGH O·RUGLQH GL G VL VXSSRQH VHPSOLFHPHQWH

FKH

O·HVLVWHQ]D GL XQ S VRWWRJUXSSR GL RUGLQH

p h LQIHULRUH D TXHOOR GHO VRWWRJUXSSR GL 6\ORZ RVVLD VL

VXSSRQH FKH VLD h < k

6HFRQGR 7HRUHPD GL 6\ORZ 6LD G XQ JUXSSR ILQLWR LO FXL RUGLQH FRQWLHQH LO IDWWRUH SULPR

p

7XWWL L VRWWRJUXSSL GL 6\ORZ GHO JUXSSR G UHODWLYL DOOR VWHVVR IDWWRUH 'LPRVWUD]LRQH

p VRQR WUD ORUR FRQLXJDWL

k

6LDQR S H T GXH VRWWRJUXSSL GL 6LORZ GLVWLQWL GL RUGLQH n s = nt = p $O JUXSSR G LO FXL RUGLQH k

ULVXOWD TXLQGL HVSULPLELOH QHOOD IRUPD n g = np FRQ n QRQ GLYLVLELOH SHU

p VL SXz DSSOLFDUH LO

WHRUHPD GL )UREHQLXV UHODWLYR D L GXH LQVLHPL S H T GD FXL VHJXH l

•

G = ∪ Sg i T FRQ g i ∈ G

•

l n s nt pk =¦ n g = np = ¦ i =1 ni i =1 ni

i=i

k

( )

l

2

GRYH

ni q

O·RUGLQH

GHO

VRWWRJUXSSR

Di = ( g i−1 Sg i ) ∩ T 'DOOD UHOD]LRQH UHODWLYD DJOL RUGLQL ULSRUWDWD QHO VHFRQGR SXQWR VL GHGXFH FKH l

n=¦ i =1

2UD RVVHUYDQGR FKH •

pk ni

•

( g i−1 Sg i ) ∩ T q XQ VRWWRJUXSSR VLD GL T VLD

•

ULFRUGDQGR FKH ( g i Sg i ) ≈ S FRPH GLPRVWUDWR QHO OHPPD ULSRUWDWR QHO SDUDJUDIR

−1

UHODWLYR DOOD h

QHFHVVDULDPHQWH ni GHYH HVVHUH GHOOD IRUPD ni = p FRQ h ≤ k SHU i = 1..l

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

l

(VVHQGR n QRQ GLYLVLELOH SHU

p ULFRUGDQGR FKH n = ¦ i =1

YDORUH

j = 1..l GHOO·LQGLFH i WDOH FKH

pk QHFHVVDULDPHQWH GHYH HVLVWHUH XQ ni

k

p = 1 GD FXL nj n j = p k

$OORUD QHFHVVDULDPHQWH •

D j ≡ T

•

D j ≡ g i−1 Sg i

4XLQGL

T = g i−1 Sg i RVVLD L S VRWWRJUXSSL GL 6\ORZ S H T ULVXOWDQR FRQLXJDWL

&ULWHULR GL XQLFLWj

,O SUHFHGHQWH WHRUHPD Gj RULJLQH DO VHJXHQWH FULWHULR G XQLFLWj GHL S VRWWRJUXSSL GL 6\ORZ L VRWWRJUXSSL GL 6\ORZ UHODWLYL DG XQR VWHVVR QXPHUR SULPR p VRQR XQLFL VH H VROR VH VRQR QRUPDOL ,QIDWWL ULFRUGDQGR FKH L S VRWWRJUXSSR GL 6\ORZ VRQR FRQLXJDWL VHJXH GDO WHRUHPD SUHFHGHQWH

T = g i−1 Sg i GD FXL VH S G S ≡ T H YLFHYHUVD VH S ≡ T S = g i−1 Sg i S G 7HU]R 7HRUHPD GL 6\ORZ 'HWWR s LO QXPHUR GL S VRWWRJUXSSL GL 6\ORZ GL XQ JUXSSR G UHODWLYL DOOR VWHVVR QXPHUR SULPR VL KD s GLYLGH Ord (G) • •

p

s = 1 mod( p) RVVLD s = np + 1 FRQ n LQWHUR

'LPRVWULDPR LO WHRUHPD

3ULPD SDUWH s GLYLGH Ord (G )

$EELDPR GLPRVWUDWR FKH WXWWL L S VRWWRJUXSSL VRQR FRQLXJDWL SHUWDQWR LO ORUR QXPHUR s q SDUL DOO·LQGLFH LQ G GHO QRUPDOL]]DWH N (S ) GRYH S q XQ S VRWWRJUXSSR GL 6\ORZ 3HUWDQWR s s =

Ord (G ) H FLz FRQFOXGH OD GLPRVWUD]LRQH Ord [ N ( S )]

6HFRQGD SDUWH s

= 1 mod( p)

6LD • •

S XQ S VRWWRJUXSSR GL 6\ORZ GL XQ JUXSSR G GL RUGLQH Ord ( S ) = p m N (S ) LO QRUPDOL]]DQWH GL S GL RUGLQH Ord [ N ( S )] = n s

$SSOLFDQGR LO WHRUHPD GL )UREHQLXV VL KD •

r

G = ∪ Sg i N (S ) FRQ g i ∈ G i=i

r

•

Ord (G ) = ¦ i =1

p m ns −1 GRYH ni = Ord (Ti ) H Ti = g i Sg i ∩ N ( S ) ni 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR 6H k LQGLFD LO QXPHUR GL S VRWWRJUXSSL GL 6\ORZ GL G DEELDPR GLPRVWUDWR QHO SDVVR GHO SUHVHQWH WHRUHPD FKH Ord (G ) = n s k SHUWDQWR SRVVLDPR VFULYHUH DSSOLFDQGR OD UHOD]LRQH VRSUD HYLGHQ]LDWH QHO VHFRQGR SXQWR FKH r

k =¦ i =1

pm ni

2UD GLYLGLDPR L FRQWULEXWL GHOOD VRPPDWRULD LQ GXH SDUWL • OD SULPD q TXHOOD FKH GHULYD GD XQ g i ∈ N (S ) LQ TXHVWR FDVR SRLFKp

g Sg i−1 i

Ti =

∩ N (S ) = S ∩ N (S ) = S

S N (S ) VHJXH

GD

FXL

uguale a S poichè g∈N ( S )

ni = Ord (Ti ) = Ord ( S ) = p m •

OD VHFRQGD q TXHOOD GHWHUPLQDWD GDJOL HOHPHQWL g i ∉ N (S ) LQ TXHVWR FDVR

g i Sg i−1 = Fi ≠ S LQ Ti =

g i Sg i−1

TXDQWR

g i ∉ N (S )

GD

FXL

VHJXH

FKH

∩ N ( S ) = Fi ∩ N ( S )

,Q EDVH D TXDQWR VRSUD RVVHUYDWR VL KD r

k =¦ i =1

pm pm = ni Ord (Tr )

r −1

pm i =1 Ord ( Fi ∩ N ( S ) g i ∉N ( S )

+¦

ORd ( S ) = p m g ∈N ( S ) r

6L RVVHUYL RUD TXDQWR VHJXH • OD SULPD SDUWH GHOO·HVSUHVVLRQH VRSUD ULSRUWDWD TXHOOD UHODWLYD DO FRQWULEXWR GHOO·HOHPHQWR g i ∈ N (S ) o o

q UDSSUHVHQWDWD GDO SULPR WHUPLQH GHO VHFRQGR PHPEUR QHOOR VYLOXSSR GL G VHFRQGR LO WHRUHPD GL )UREHQLXV HVLVWH DOPHQR XQ HOHPHQWR g i ∈ N (S ) SRLFKp O·HOHPHQWR QHXWUR e ∈ N (S )

o

q FRVWLWXLWD GD XQ VROR DGGHQGR GHOOD VRPPDWRULD LQ TXDQWR RJQL g i ∈ N (S ) q −1

FRQWHQXWR LQ g i Sg i •

= S VWDQWH LO IDWWR FKH S N (S ) H TXLQGL QRQ q ULGXWWLYR

DYHU VXSSRVWR FKH WDOH HOHPHQWR VLD O·XOWLPR GHOOD VRPPDWRULD OD VHFRQGD SDUWH TXHOOD UHODWLYD DO FRQWULEXWR GHJOL HOHPHQWL g i ∉ N (S ) q LQYHFH UDSSUHVHQWDWD GDOOD VRPPDWRULD

r −1

pm

i =1

i

¦ Ord ( F ∩ N ( S )

LQ FXL SHU RJQL DGGHQGR VL KD g i ∉N ( S )

FKH o o

Ti = g i Sg i−1 ∩ N ( S ) = Fi ∩ N ( S ) ULVXOWD HVVHUH XQ VRWWRJUXSSR GL Fi Fi q LVRPRUIR D S FRQIURQWD TXDQWR GLPRVWUDWR QHO SDUDJUDIR UHODWLYR DOOD m

VFRPSRVL]LRQH GL XQ JUXSSR LQ VRWWRJUXSSL H SHUWDQWR Ord ( Fi ) = p l

o

GDO SXQWR SUHFHGHQWH VL GHGXFH FKH Ord (Ti ) = p i FRQ li ≤ m LQ TXDQWR O·RUGLQH GL Ti GHYH GLYLGHUH TXHOOR GL Fi

o

QHFHVVDULDPHQWH GHYH HVVHUH li < m RVVLD Ti GHYH HVVHUH XQ VRWWRJUXSSR SURSULR GL Fi LQIDWWL N (S ) HVVHQGR XQ VRWWRJUXSSR GL G H FRQWHQHQGR LO S VRWWRJUXSSR GL 6\ORZ S GL RUGLQH GL

RUGLQH

p m SXz HVVR VWHVVR FRQWHQHUH S VRWWRJUXSSL GL 6\ORZ DO SL

p m

VXSSRQLDPR

3DJ

RUD

SHU

DVVXUGR

FKH

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

Ord (Ti ) = Ord [ Fi ∩ N ( S )] = p m FRPH FRQVHJXHQ]D SRLFKp Ti GHYH HVVHUH m

VRWWRJUXSSR GL Fi H SRLFKp Ord ( Fi ) = p LQ TXDQWR S ≈ Fi VL KD FKH

Ti = Fi = g i Sg i−1 RUD Ti GHYH HVVHUH DQFKH XQ VRWWRJUXSSR GL N (S ) HG HVVHQGR GL RUGLQH

p m ULVXOWD XQ S VRWWRJUXSSR GL 6\ORZ GL N (S ) VLFFRPH S N (S )

N (S ) FRQWLHQH XQ VROR S VRWWRJUXSSR GL 6\ORZ FRQIURQWD LO FULWHULR GL XQLFLWj −1

SUHFHGHQWHPHQWH GLPRVWUDWR H SHUWDQWR GHYH HVVHUH Ti = Fi = g i Sg i

= S RVVLD

g i ∈ N (S ) FRQWUR O·LSRWHVL g i ∉ N (S ) GD FXL VHJXH FKH QRQ SXz HVVHUH Ord (Ti ) = Ord [ Fi ∩ N ( S )] = p m 8WLOL]]DQGR OH RVVHUYD]LRQL ULSRUWDWH QHL SXQWL SUHFHGHQWL VL SXz VYLOXSSDUH O·HVSUHVVLRQH GL k FRPH VHJXH r

k=¦ i =1

pm pm = ni Ord (Ti )

r −1

pm +¦ = i =1 Ord ( Fi ∩ N ( S ) g i ∉N ( S )

ORd ( S ) = p m g ∈N ( S ) i

r −1

pm

i =1

p li

= 1+ ¦

r −1

g i ∉N ( S )

= 1 + ¦ p m −li g i ∉N ( S ) i =1

r −1 ª r −1 º k = 1 + ¦ p m−li g i ∉N ( S ) = 1 + «¦ p m−li −1 g i∉N ( S ) » p = 1 + np RVVLD k = 1 mod( p) i =1 1 ¬ i =

¼ n

H FLz FRPSOHWD OD GLPRVWUD]LRQH

(VHPSL 6XSSRQLDPR FKH Ord (G ) •

= 34 5 2 DOORUD VL KD 4

XQ VROR VRWWRJUXSSR GL 6\ORZ H 1 GL RUGLQH 3 SRLFKp HVVHUH GLYLVLELOH SHU Ord (G )

k = 1 mod(3) = 3s + 1 GHYH

4 2

= 3 5 q FLz VL KD VROR SHU s = 0 3

2

•

VRWWRJUXSSL H 1,1 H 1, 2 H 1,3 GL RUGLQH ULVSHWWLYDPHQWH 3 3 3

•

XQ VROR VRWWRJUXSSR GL 6\ORZ H 2 GL RUGLQH 3 SRLFKp

4

= 3 5 q FLz VL KD VROR SHU s = 0 VRWWRJUXSSL H 2,1 H 2, 2 H 2,3 GL RUGLQH ULVSHWWLYDPHQWH 5 HVVHUH GLYLVLELOH SHU Ord (G )

•

k = 1 mod(5) = 5s + 1 GHYH

4 2

*UXSSL ULVROXELOL 8Q JUXSSR G q GHWWR ULVROXELOH VH VL YHULILFDQR OH VHJXHQWL WUH FRQGL]LRQL HVLVWH XQD FDWHQD GL VRWWRJUXSSL GL G FRQILJXUDWD FRPH VHJXH G ≡ Gn ⊃ Gn−1 ⊃ ..... ⊃ G1 ⊃ G0 ≡ {e} GRYH {e}LQGLFD LO VRWWRJUXSSR FKH FRQWLHQH LO

VROR HOHPHQWR QHXWUR Gk −1 Gk SHU k = 1..n

Gk / Gk −1 q DEHOLDQR SHU k = 1..n

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

2VVHUYD]LRQH /D GHILQL]LRQH GL JUXSSR ULVROXELOH q FHQWUDOH QHOOD WHRULD GL *DORLV VXOOH FRQGL]LRQL GL ULVROYLELOLWj SHU UDGLFDOL GHOOH HTXD]LRQL DOJHEULFKH LQ ULIHULPHQWR DG XQ SDUWLFRODUH WLSR GL JUXSSR GHQRPLQDWR JUXSSL VLPPHWULFR $ TXHVWR IDWWR VL GHYH OD VFHOWD GHO QRPH GL ´JUXSSR ULVROXELOHµ XWLOL]]DWR QHOOD GHILQL]LRQH

3ULPR WHRUHPD VXL JUXSSL ULVROXELOL 6H G q XQ JUXSSR ULVROXELOH DOORUD RJQL VRWWRJUXSSR H GL G ULVXOWD ULVROXELOH %LVRJQD GLPRVWUDUH FKH YDOJRQR OH WUH FRQGL]LRQL GL ULVROXELOLWj SHU H (VLVWHQ]D GHOOD FDWHQD GL VRWWRJUXSSL H ≡ H n ⊃ H n −1 ⊃ ..... ⊃ H 1 ⊃ H 0 ≡ {e} ,QIDWWL SRVWR

H k = H ∩ G k SHU k = 0..n

VL KD SHU k = n G k = G n = G H n = H ∩ G n = H ∩ G = H

k = (n − 1) H n −1 = H ∩ G n−1 ⊂ H n ≡ H H n−1 ULVXOWD HVVHUH VRWWRJUXSSR GL G LQ TXDQWR LQWHUVH]LRQH H H G n −1 FKH VRQR VRWWRJUXSSL GL G H n −1 q TXLQGL DQFKH VRWWRJUXSSR GL H LQ TXDQWR H n −1 ⊂ H n ≡ H SHU LO JHQHULFR YDORUH GL k = 1..(n − 2) YDOH XQ UDJLRQDPHQWR DQDORJR HVVHQGR H k = H ∩ G k ⊂ H k +1 DOORUD H k q VRWWRJUXSSR GL H k +1 HVVHQGR VRWWRJUXSSR GL G HG HVVHQGR H k ⊂ H k +1 SHU k = 0 G k = G0 = {e} H 0 = H ∩ G0 = H ∩ {e} = {e} 1RUPDOLWj GHL VRWWRJUXSSL GHOOD FDWHQD H k −1 H k SHU k = 1..n ,QIDWWL FRQVLGHULDPR XQ h ∈ H k H u ∈ H k −1 SHU GHILQL]LRQH h ∈ H H h ∈ Gk u ∈ H H SHU

u ∈ G k −1 HVVHQGR G k −1 G k huh −1 ∈ G k −1 H HVVHQGR H VRWWRJUXSSR GL G huh −1 ∈ H $OORUD huh −1 ∈ H ∩ Gk −1 = H k −1 GD FXL VHJXH H k −1 H k $EHOLDQLWj GHL JUXSSL TXR]LHQWL H k / H k −1 q DEHOLDQR SHU k = 1..n 3HU GLPRVWUDUH WDOH SURSULHWj RFFRUUH IDUH ULIHULPHQWR DO SULPR WHRUHPD GHJOL LVRPRUILVPL RVVHUYDQGR LQQDQ]L WXWWR FKH H k = H ∩ G k

H k −1 = H ∩ Gk −1 = ( H ∩ G k ) ∩ G k −1 FLz q GRYXWR DO IDWWR FKH Gk −1 ⊂ Gk $OORUD DSSOLFDQGR LO SULPR WHRUHPD VXJOL LVRPRUILVPL VHJXH

H k / H k −1 = H ∩ Gk /[( H ∩ G k ) ∩ Gk −1 ] ≈ G k −1 ( H ∩ G k ) / G k −1 H k / H k −1 ≈ Gk −1 ( H ∩ Gk ) / G k −1

2UD G k −1 ( H ∩ Gk ) / G k −1 q XQ VRWWRJUXSSR GHO JUXSSR DEHOLDQR SHU LSRWHVL G k / G k −1 LQ TXDQWR G k −1 ( H ∩ G k ) q XQ VRWWRJUXSSR GL Gk FRPH GLPRVWUDWR DO SDVVR SULPR GHOOD GLPRVWUD]LRQH GHO SULPR WHRUHPD GHJOL LVRPRUILVPL H Gk −1 ⊂ Gk −1 ( H ∩ G k ) 3HUWDQWR G k −1 ( H ∩ Gk ) / G k −1 ULVXOWD DEHOLDQR $OORUD VL SXz FRQFOXGHUH FKH DQFKH H k / H k −1 q DEHOLDQR LQ TXDQWR q LVRPRUIR GL XQ JUXSSR DEHOLDQR

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

6HFRQGR WHRUHPD VXL JUXSSL ULVROXELOL 6H N G H VH N H G / N VRQR ULVROXELOL DOORUD G q ULVROXELOH (VVHQGR N QRUPDOH H ULVROXELOH VHJXH OD YDOLGLWj SHU N GHOOH FRQGL]LRQL GL ULVROXELOLWj H SHUWDQWR • HVLVWH XQD FDWHQD GL VRWWRJUXSSL GL N FRQILJXUDWD FRPH VHJXH N ≡ N m ⊃ N m −1 ⊃ ..... ⊃ N1 ⊃ N 0 ≡ {e} •

N k −1 N k SHU k = 1..m

•

N k / N k −1 q DEHOLDQR SHU k = 1..m

6L RVVHUYL RUD FKH L JUXSSL N k SHU k = 1..m VRQR DQFKH VRWWRJUXSSL QRUPDOL GHO JUXSSR G H SHUWDQWR IDQQR SDUWH GL XQ SH]]R GHOOD FDWHQD GL VRWWRJUXSSL UHODWLYL DOOD ULVROXELOLWj GL G 3HU WDOH PRWLYR LQGLFKLDPR N k DQFKH FRQ LO VLPEROR Gk SHU k = 1..m SHU LQFLVR ULVXOWD

N = N m = G m &RQVLGHULDPR RUD LO IDWWR FKH SHU LSRWHVL DQFKH G / N q ULVROXELOH H SHUWDQWR YDOJRQR OH WUH FRQGL]LRQL GL ULVROXELOLWj GL VHJXLWR ULSRUWDWH • HVLVWHQ]D GHOOD FDWHQD GL VRWWRJUXSSL GL G / N FRQILJXUDWD FRPH VHJXH G / N ≡ H n ⊃ H n−1 ⊃ ..... ⊃ H 1 ⊃ H 0 ≡ N / N = N •

H k −1 H k SHU k = 1..n

•

H k / H k −1 q DEHOLDQR SHU k = 1..n

$QDOL]]LDPR OD QDWXUD GHL JUXSSL H k 7DOL JUXSSL VRQR GHL VRWWRJUXSSL GHO JUXSSR TXR]LHQWH

G / N RVVLD L ORUR HOHPHQWL VRQR XQ VRWWRLQVLHPH GL ODWHUDOL GL G / N ,Q PRGR SL HVSOLFLWR VH Ord (G ) LQGLFKLDPR FRQ s = O·LQGLFH GL N LQ G VL KD Ord ( N ) G / N = {eN , g1 N , g 2 N , g 3 N ,....., g s N } FRQ g i ∈ G SHU i = 1..s

{

}

H k = eN , g k1 N , g k 2 N , gk 3 N ,....., g kj N GRYH H k

q XQ VRWWRLQVLHPH GHL ODWHUDOL GL N FKH YHULILFDQR OH LSRWHVL GL JUXSSR

&RPH FDVR SDUWLFRODUH HYLGHQ]LDPR FKH H 0 q FRVWLWXLWR GDO VROR HOHPHQWR QHXWUR H YDOH

H 0 = {eN } = N = Gm

,QGLFDQGR RUD FRQ

{

}

Gm+ k = e, g k1 , g k 2 , gk 3 ,....., g kj ⊂ G SHU k = 1..n LO VRWWRLQVLHPH GL HOHPHQWL GL G DWWUDYHUVR L TXDOL VL JHQHUDQR L ODWHUDOL GL H k ,Q SUHFHGHQ]D FRQIURQWD LO SDUDJUDIR UHODWLYR DL JUXSSL TXR]LHQWL DEELDPR GLPRVWUDWR FKH H k G / N Gm+ k G H SL LQ JHQHUDOH

H k −1 H k Gm+ ( k −1) Gm+ k SHU k = 1..n

HG q EHQ SRVWD OD VHJXHQWH QRWD]LRQH

H k = G m + k / N SHU k = 1..n

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² 6WUXWWXUD $OJHEULFD GL *UXSSR

$ TXHVWR SXQWR SHU OD UHOD]LRQH H k / H k −1 ≡

Gm + k / N G m* +( k −1) / N

VLDPR LQ FRQGL]LRQH GL DSSOLFDUH

LO VHFRQGR WHRUHPD GHOO·LVRPRUILVPR HVVHQGR N ≡ Gm ⊂ G m + k

Gm+ ( k −1) Gm+ k LQ PRGR GD

RWWHQHUH

H k / H k −1 ≡

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3ULPD GL GLPRVWUDUH LO OHPPD HYLGHQ]LDPRQH LO VLJQLILFDWR HVVR DIIHUPD FKH SHU n ≥ 5 QRQ SXz HVLVWHUH XQ VRWWRJUXSSR QRUPDOH GHO JUXSSR DOWHUQR FKH DEELD VROR SHUPXWD]LRQL FKH PXRYRQR WXWWL JOL HOHPHQWL GHOO·LQVLHPH {1,2,...n} RVVLD DG HVHPSLR QHO FDVR n = 5 DEEDL VROR FLFOL GL RUGLQH FLQTXH VL YHGD QHL VXFFHVVLYL SDUDJUDIL OD WDEHOOD FKH LOOXVWUD WXWWH OH SHUPXWD]LRQL GL A5 'LPRVWULDPR RUD LO OHPPD 6LD ρ XQD SHUPXWD]LRQH DSSDUWHQHQWH D N FKH DJLVFD VX XQ HOHPHQWR a ∈ {1,2,...n} WDOH SHUPXWD]LRQH GHYH HVLVWHUH SHU LSRWHVL LQ TXDQWR N q XQ VRWWRJUXSSR QRQ EDQDOH &DVR aρ = a 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL

ρ q OD SHUPXWD]LRQH FHUFDWD H OD GLPRVWUD]LRQH q FRQFOXVD &DVR ρ

2

≠ I SHUPXWD]LRQH XQLWj

$OORUD VLD KD aρ LQ TXDQWR ρ

= b H bρ = aρ 2 = c GRYH a ≠ b DOWULPHQWL FL WURYHUHPPR QHO FDVR a ≠ c

2

≠ I 6H b = c VHJXH bρ = b H GXQTXH ρ q OD SHUPXWD]LRQH FHUFDWD H OD GLPRVWUD]LRQH q FRQFOXVD 6H b ≠ c VL VFHOJDQR DOWUL GXH HOHPHQWL d HG e DSSDUWHQHQWL D {1,2,...n} H GLVWLQWL GDL SUHFHGHQWL WDOL HOHPHQWL HVLVWRQR LQ TXDQWR n ≥ 5 H VL YDOXWL DO VHJXHQWH SHUPXWD]LRQH π = (cde) −1 ρ (cde) 9DOH TXDQWR VHJXH • π ∈ An RVVLD q SHUPXWD]LRQH SDUL LQ TXDQWR q LO SURGRWWR GL SHUPXWD]LRQL SDUL • •

π ∈ N LQ TXDQWR N An aπ = aρ = b bπ = bρ (cde) = d

• 4XLQGL SRVVLDPR GHGXUUH FKH

π ≠ ρ πρ −1 ≠ I H πρ −1 ∈ N SHU OD SURSULHWj GL FKLXVXUD LQ TXDQWR HQWUDPEH DSSDUWHQJRQR D N

6L YDOXWL RUD O·D]LRQH πρ

−1

LQ a

aπρ −1 = bρ −1 = a 3HUWDQWR OD σ = &DVR ρ

πρ −1 q OD SHUPXWD]LRQH FHUFDWD H OD GLPRVWUD]LRQH q FRQFOXVD

2

= I SHUPXWD]LRQH XQLWj $OORUD VLD KD aρ = b GRYH a ≠ b DOWULPHQWL FL WURYHUHPPR QHO FDVR

6L RVVHUYL FKH OD WUDVSRVL]LRQH τ

= (ab) q WDOH FKH aτ = b H τ 2 = I SHUz ρ QRQ SXz FRLQFLGHUH FRQ τ LQ TXDQWR ρ ∈ An q XQD SHUPXWD]LRQH SDUL D GLIIHUHQ]D GL τ $OORUD QHFHVVDULDPHQWH ρ GHYH DJLUH VX XQ WHU]R HOHPHQWR c GLYHUVR GD a H b 6H cρ = c ρ q OD SHUPXWD]LRQH FHUFDWD H OD GLPRVWUD]LRQH q FRQFOXVD 6H cρ = d ≠ c VL VFHOJD XQ DOWUR HOHPHQWR e GLVWLQWR GL SUHFHGHQWL H VL DQDOL]]L DO SHUPXWD]LRQH

π = (cde) −1 ρ (cde) 3URVHJXHQGR FRPH QHO FDVR GXH VL FRQFOXGH OD GLPRVWUD]LRQH

OHPPD

6H N Σ n DOORUD N ⊂ An RSSXUH N ∩ An q WDOH FKH Ord ( N ) / Ord ( N ∩ An ) = 2 RVVLD

N ∩ An VRWWRJUXSSR GL N GL LQGLFH GXH ,QIDWWL DSSOLFDQGR OH SURSULHWj UHODWLYH DOOD QRUPDOLWj GHOO·LQWHUVH]LRQH WUD VRWWRJUXSSL HG L OHPPL VXL JUXSSL QRUPDOL H TXR]LHQWL HVVHQGR N Σ n H An Σ n VL KD •

N ∩ An An

•

NAn / An VRWWRJUXSSR GL Σ n / An GD FXL VHJXH FKH Ord ( NAn / An ) q SDUL DG XQR RSSXUH

D GXH $SSOLFDQGR LO WHRUHPD VXJOL LVRPRUILVPL VHJXH

NAn / An ≈ N /( N ∩ An ) 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL (VVHQGR N /( N ∩ An ) LVRPRUIR NAn / An VHJXH FKH DQFKH N /( N ∩ An ) KD XQR R GXH HOHPHQWL •

VH Ord ( NAn / An ) = 1 VHJXH NAn / An = {An } GD FXL N ⊂ An

•

Ord [ N /( N ∩ An )] = 2 VHJXH Ord ( N ) / Ord ( N ∩ An ) = 2

*UXSSR VLPPHWULFR GL RUGLQH

,Q TXHVWR SDUDJUDIR YRJOLDPR DQDOL]]DUH QHO GHWWDJOLR OD VWUXWWXUD GL Σ 2 6DSSLDPR FKH LO JUXSSR VLPPHWULFR GL RUGLQH KD XQ QXPHUR GL HOHPHQWL SDUL D Ord (Σ 2 ) = 2! = 2 3HU HVDWWH]]D JOL HOHPHQWL VRQR GDWL GDOOH VHJXHQWL SHUPXWD]LRQL Σ 2 ≡ {σ e , σ 1 }

§1 2 ·

¸¸ σ e = ¨¨ ©1 2 ¹

LQGHQWLWj σ 1

§1 2· ¸¸ = ¨¨ ©2 1¹

Σ 2 q GXQTXH XQ JUXSSR SULPR SHUWDQWR ULVXOWD FLFOLFR H TXLQGL DEHOLDQR ,O JUXSSR DOWHUQR A2 KD XQ VROR HOHPHQWR H FRLQFLGH FRQ OD SHUPXWD]LRQH LGHQWLWj A2 = {σ e } = {e} 5LVROXELOLWj GHO JUXSSR Σ 2

'D WXWWR TXDQWR SUHFHGH SRVVLDPR GHGXUUH FKH LO JUXSSR VLPPHWULFR GL RUGLQH ULVXOWD ULVROXELOH LQ TXDQWR • (VLVWH OD FDWHQD GL VRWWRJUXSSL Σ 2 ⊃ A2 = {e}

A2 Σ 2 Σ 3 / A2 ≡Σ 3 /{e} =Σ 3 DEHOLDQR

• •

2VVHUYD]LRQH

/D ULVROXELOWj GHO JUXSSR Σ 2 q FRUUHODWD DOO·HVLVWHQ]D GL XQD IRUPXOD ULVROXWLYD SHU UDGLFDOL GHOOH HTXD]LRQL DOJHEULFKH GL VHFRQGR JUDGR

*UXSSR VLPPHWULFR GL RUGLQH

,Q TXHVWR SDUDJUDIR YRJOLDPR DQDOL]]DUH QHO GHWWDJOLR OD VWUXWWXUD GL Σ 3 6DSSLDPR FKH LO JUXSSR VLPPHWULFR GL RUGLQH KD XQ QXPHUR GL HOHPHQWL SDUL D Ord (Σ 3 ) 3HU HVDWWH]]D JOL HOHPHQWL VRQR GDWL GDOOH VHJXHQWL SHUPXWD]LRQL Σ 3 ≡ {σ e , σ 1 , σ 2 , σ 3 , σ 4 , σ 5 }

§1 2 3 ·

¸¸ = (23) σ 1 = ¨¨ ©1 3 2 ¹ § 1 2 3·

¸¸ = (12) σ 3 = ¨¨ © 2 1 3¹ § 1 2 3·

¸¸ = (13) σ 5 = ¨¨ ©3 2 1¹

§ 1 2 3·

¸¸ = (123) = (12)(13) σ 2 = ¨¨ © 2 3 1¹

¸¸ = (132) = (13)(12) σ 4 = ¨¨ ©3 1 2¹

¸¸ σ e = ¨¨ ©1 2 3 ¹

§1 2 3·

§1 2 3 ·

LGHQWLWj

,O JUXSSR DOWHUQR A3 KD Ord ( A3 ) = 3! / 2 = 3 HOHPHQWL GL VHJXLWR ULSRUWDWL 3DJ

= 3!= 6

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A3 ≡ {σ e , σ 2 , σ 4 ,} = {σ e , (12)(13), (13)(12)} = {σ e , (123), (132)} 9DOJRQR OH VHJXHQWL SURSULHWj SHU A3 •

q XQ VRWWRJUXSSR QRUPDOH GL Σ 3 FRPH GLPRVWUDWR QHO FDVR JHQHUDOH

•

RJQL HOHPHQWR SXz HVVHUH SRVWR QHOOD IRUPD GL XQ FLFOR LQ FRHUHQ]D FRQ LO IDWWR FKH RJQL FRSSLD GL WUDVSRVL]LRQL VL SXz HVSULPHUH FRPH XQ FLFOR q XQ JUXSSR SULPR SHUWDQWR ULVXOWD • FLFOLFR HG DEHOLDQR • O·HOHPHQWR JHQHUDWRUH q GDWR GDO FLFOR (123) RVVLD A3 q JHQHUDWR GD XQ XQLFR

•

= (132) (123) 3 = (132)(123) = σ e q VHPSOLFH LQIDWWL L SRVVLELOL VRWWRJUXSSL GL A3 VRQR {σ e , (123)} H {σ e , (132)} PD LQ FLFOR ,QIDWWL (123)

•

2

HQWUDPEL L FDVL QRQ YDOH DG HVHPSLR OD SURSULHWj GL FKLXVXUD QHO SULPR FDVR

(123) 2 = (132)

PHQWUH

QHO

VHFRQGR

FDVR

(132)(132) = (123) 4 = (123) 3 (123) = σ e (123) = (123) 3HU TXDQWR ULJXDUGD LO JUXSSR Σ 3 / A3 YDOJRQR OH VHJXHQWL SURSULHWj •

KD GXH VROL HOHPHQWL Σ 3 / A3 = {A3 , D3 } FRQ D3 = {σ 1 , σ 3 , σ 5 } = {( 23), (12), (13)}

•

q SULPR H TXLQGL ULVXOWD DQFKH FLFOLFR H DEHOLDQR FRQ HOHPHQWR JHQHUDWRUH D3 ( D3 ) = A3

•

q VHPSOLFH

2

5LVROXELOLWj GHO JUXSSR Σ 3

'D WXWWR TXDQWR SUHFHGH SRVVLDPR GHGXUUH FKH LO JUXSSR VLPPHWULFR GL RUGLQH ULVXOWD ULVROXELOH LQ TXDQWR • (VLVWH OD FDWHQD GL VRWWRJUXSSL Σ 3 ⊃ A3 ⊃ {e} • •

A3 Σ 3 {e} A3

Σ 3 / A3 A3 /{e} = A3 VRQR DEHOLDQL 6RWWRJUXSSL QRUPDOL GHO JUXSSR Σ 3

9RJOLDPR YHULILFDUH FKH O·XQLFR VRWWRJUXSSR QRUPDOH SURSULR GL

Σ 3 q LO JUXSSR DOWHUQR A3

,QIDWWL VXSSRQLDPR FKH HVLVWD XQ VRWWRJUXSSR QRUPDOH N ≠ A3 $OORUD VL SRVVRQR YHULILFDUH LO VHJXHQWL FDVL N ⊂ A3 FLz LPSOLFD FKH DG N DSSDUWLHQH DOPHQR XQD SHUPXWD]LRQH SDUL σ 2 σ 4 • GLYHUVD GDOOD SHUPXWD]LRQL LGHQWLWj H TXLQGL VHJXH FKH N = A3 LQ TXDQWR XQD GHOOH GXH SHUPXWD]LRQL SDUL q VXIILFLHQWH D JHQHUDUH WXWWR A3 •

N ∩ A3 ≠ {0} GRYH {0} LQGLFD O·LQVLHPH YXRWR H N ∩ A3 ≠ {e} LQ TXHVWR FDVR N QRQ

FRLQFLGH FRQ LO JUXSSR DOWHUQR PD FRQWLHQH DOPHQR XQ SHUPXWD]LRQH SDUL HG KD HOHPHQWL GL D3 SHU TXDQWR RVVHUYDWR DO SUHFHGHQWH SXQWR A3 YLHQH JHQHUDWR GDOOD SHUPXWD]LRQH SDUL DSSDUWHQHQWH D N H SHUWDQWR VL KD FKH A3 ⊂ N FDVR LPSRVVLELOH LQ TXDQWR A3 q •

PDVVLPDOH H QRQ SXz HVVHUH FRQWHQXWR LQ DOFXQ VRWWRJUXSSR QRUPDOH N ∩ A3 = {e} LQ TXHVWR FDVR N QRQ FRQWLHQH SHUPXWD]LRQL SDUL DG HVFOXVLRQH GHOOD SHUPXWD]LRQH LGHQWLWj RVVLD FRQWLHQH VROR HOHPHQWL GL ROWUH O·LGHQWLWj D3 L FDVL SRVVLELOL

{e, (23)} {e, (12)} {e, (13)}FKH

UDSSUHVHQWDQR GHL VRWWRJUXSSL PHQWUH VL HVFOXGH 3DJ

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{e, (23), (12)} LQ

TXDQWR QRQ q XQ VRWWRJUXSSR SRLFKp LO SURGRWWR

(23)(12) q XQD

SHUPXWD]LRQH SDUL H QRQ SXz DSSDUWHQHUH D N RUD q IDFLOH YHULILFDUH SHU FDOFROR GLUHWWR FKH L WUH VRWWRJUXSSL {e, ( 23)} {e, (12)} {e, (13)}QRQ VRQR QRUPDOL

1RQ VL SRVVRQR YHULILFDUH DOWUL FDVL H SHUWDQWR QRQ SXz HVLVWHUH XQ VRWWRJUXSSR QRUPDOH N ≠ A3

2VVHUYD]LRQH

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HTXD]LRQL DOJHEULFKH GL WHU]R JUDGR

*UXSSR VLPPHWULFR GL RUGLQH

,Q TXHVWR SDUDJUDIR YRJOLDPR DQDOL]]DUH QHO GHWWDJOLR OD VWUXWWXUD GL Σ 4 6DSSLDPR FKH LO JUXSSR VLPPHWULFR GL RUGLQH KD XQ QXPHUR GL HOHPHQWL SDUL D Ord (Σ 4 ) = 4! = 24 3HU HVDWWH]]D JOL HOHPHQWL VRQR GDWL GDOOH VHJXHQWL SHUPXWD]LRQL

­σ e , σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 , σ 7 , σ 8 , σ 9 , σ 10 , σ 11 , σ 12 ,½ Σ4 ≡ ® ¾ ¯σ 13 , σ 14 , σ 15 , σ 16 , σ 17 , σ 18 , σ 19 , σ 20 , σ 21 , σ 22 , σ 23 ¿

§1 2 3 4 ·

§1 = ¨¨ ©1 4· §1 ¸¸ = (24) σ 7 = ¨¨ 2¹ ©2 4· §1 ¸¸ = (1234) = (12)(13)(14) σ 11 = ¨¨ 1¹ ©2

¸¸ = (34) σ 1 = ¨¨ ©1 2 4 3 ¹ §1 2 3

σ 5 = ¨¨ ©1 4 3 §1 2 3

σ 9 = ¨¨ ©2 3 4

σ 3

2 3 4· ¸ = (23) 3 2 4 ¸¹ 2 3 4· ¸ = (12) 1 3 4 ¸¹ 2 3 4· ¸ = (1243) = (12)(14)(13) 4 1 3 ¸¹

§1 2 3 4·

§1 2 3 4·

§1 2 3 4·

§ 1 2 3 4·

¸¸ = (1342) = (13)(14)(12) σ 15 = ¨¨ ¸¸ = (13) σ 13 = ¨¨ ©3 1 4 2¹ ©3 2 1 4¹ ¸¸ = (1324) = (13)(12)(14) σ 19 = ¨¨ ¸¸ = (1432) = (14)(13)(12) σ 17 = ¨¨ ©3 4 2 1¹ © 4 1 2 3¹ §1 2 3 4·

¸¸ = (14) σ 21 = ¨¨ ©4 2 3 1¹

§ 1 2 3 4·

¸¸ = (1423) = (14)(12)(13) σ 23 = ¨¨ © 4 3 1 2¹

§1 2 3 4 ·

¸¸ σ e = ¨¨ ©1 2 3 4 ¹

LGHQWLWj

§ 1 2 3 4·

¸¸ = (124) = (12)(14) σ 4 = ¨¨ ©2 4 3 1¹ §1 2 3 4·

¸¸ = (134) = (13)(14) σ 8 = ¨¨ ©3 2 4 1¹ § 1 2 3 4·

¸¸ = (143) = (14)(13) σ 12 = ¨¨ © 4 2 1 3¹

§1 = ¨¨ ©2 §1 σ 6 = ¨¨ ©3 σ 2

σ 10 σ 14

3DJ

§1 = ¨¨ ©4 §1 = ¨¨ ©1

2 3 2 1

3 1 3 2

4· ¸ = (123) = (12)(23) 4 ¸¹ 4· ¸ = (132) = (13)(12) 4 ¸¹

2 3 4· ¸ = (142) = (14)(12) 1 3 2 ¸¹ 2 3 4· ¸ = (234) = (23)(34) 3 4 2 ¸¹

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL

§1 2 3 4 ·

¸¸ = (243) = (24)(23) σ 16 = ¨¨ ©1 4 2 3 ¹

§1 2 3 4·

¸¸ = (12)(34) = (123)(143) σ 18 = ¨¨ 2 1 4 3 © ¹ §1 2 3 4·

¸¸ = (13)(24) = (132)(142) σ 20 = ¨¨ 3 4 1 2 © ¹

§1 2 3 4·

¸¸ = (14)(23) = (142)(132) σ 22 = ¨¨ 4 3 2 1 © ¹

*UXSSR DOWHUQR A4

,O JUXSSR DOWHUQR A4 KD Ord ( A3 ) = 4! / 2 = 12 HOHPHQWL GL VHJXLWR ULSRUWDWL

A4 ≡ {σ e , σ 2 , σ 4 , σ 6 , σ 8 , σ 10 , σ 12 , σ 14 , σ 16 , σ 18 , σ 20 , σ 22 } =

{σ e , (123), (124), (132), (134), (142), (143), (234), (243), (123)(143), (132)(142), (142)(132)} = ­σ e , (12)(13), (12)(14), (13)(12), (13)(14), (14)(12), (14)(13), (23)(24), (24)(23),½ ® ¾ ¯(12)(34), (13)(24), (14)(23) ¿ 9DOJRQR OH VHJXHQWL SURSULHWj SHU A4 • q XQ VRWWRJUXSSR QRUPDOH GL Σ 4 FRPH GLPRVWUDWR QHO FDVR JHQHUDOH •

RJQL HOHPHQWR SXz HVVHUH SRVWR QHOOD IRUPD GL XQ FLFOR LQ FRHUHQ]D FRQ LO IDWWR FKH RJQL FRSSLD GL WUDVSRVL]LRQL VL SXz HVSULPHUH FRPH XQ FLFOR ,QGLFDWR FRQ D4 = {σ 1 , σ 3 , σ 5 , σ 7 , σ 9 , σ 11 , σ 13 , σ 15 , σ 17 , σ 19 , σ 21 , σ 23 } O·LQVLHPH GL WXWWH OH SHUPXWD]LRQL GLVSDUL VL RVVHUYL FKH WDOH LQVLHPH QRQ FRVWLWXLVFH XQ VRWWRJUXSSR GL Σ 4 LQ TXDQWR • •

QRQ q GRWDWR GHOO·HOHPHQWR QHXWUR HG DQFKH VH VL DJJLXQJHVVH O·HOHPHQWR QHXWUR σ e QRQ YDOH OD SURSULHWj GL FKLXVXUD LQ

TXDQWR LO SURGRWWR GL GXH SHUPXWD]LRQL GLVSDUL IRUQLVFH XQD SHUPXWD]LRQH SDUL 2VVHUYLDPR LQILQH FKH JOL HOHPHQWL GL A4 VL SRVVRQR GLYHGHUH LQ GXH VRWWRLQVLHPL • LO SULPR FRVWLWXLWR GDOOH SULPH QRYH SHUPXWD]LRQL SDUL HVSULPLELOL RJQXQD FRPH XQ VLQJROR FLFOR DG HVFOXVLRQH GHOOD SHUPXWD]LRQH XQLWj • LO VHFRQGR FRVWLWXLWR GDOOH XOWLPH SHUPXWD]LRQL HVSULPLELOL FRPH GXH FLFOL H FRLQYROJHQWL WXWWL H JOL HOHPHQWL GL {1,2,3,4} ,QGLFKLDPR FRQ LQGLFKLDPR FRQ • V O·LQVLHPH GHOOH SHUPXWD]LRQL GHO SULPR WLSR HVFOXGHQGR OD SHUPXWD]LRQH XQLWj

V = {σ e , σ 2 , σ 4 , σ 6 , σ 8 , σ 10 , σ 12 , σ 14 , σ 16 }

•

V4 O·LQVLHPH GHOOH SHUPXWD]LRQL GHO VHFRQGR WLSR FRQ O·DJJLXQWD GHOOD SHUPXWD]LRQH XQLWj V4 = {σ e , σ 18 , σ 20 , σ 22 } = {σ e , (12)(34), (13)(24), (14)(23)}

&RQ WDOH QRPHQFODWXUD LO JUXSSR DOWHUQR VL SXz UDSSUHVHQWDUH VLQWHWLFDPHQWH FRPH VHJXH A4 = {V , V4 } 6L RVVHUYL LQILQH FKH YDOJRQR OH VHJXHQWL UHOD]LRQL 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL • • •

σ 18 = (123)(143) = σ 2σ 12 σ 20 = (132)(142) = σ 6σ 10 σ 22 = (142)(123) = σ 10σ 2

3HUWDQWR

V4 = {σ e , σ 18 , σ 20 , σ 22 } = {σ e , σ 2σ 12 , σ 6σ 10 , σ 10σ 2 }

RVVLD V4 q JHQHUDWR GD DOFXQL HOHPHQWL GL V H TXLQGL A4 q JHQHUDWR GDL VROL HOHPHQWL GL V RVVLD GDL FLFOL

6RWWRJUXSSL QRUPDOL GHO JUXSSR DOWHUQR A4

9RJOLDPR GHWHUPLQDUH WXWWL L VRWWRJUXSSL QRUPDOL SURSUL GL A4 $ WDOH VFRSR GLPRVWULDPR FKH V4 q XQ VRWWRJUXSSR QRPDOH DEHOLDQR GL A4

,QIDWWL RVVHUYDQGR FKH GHWWD σ ∈ V4 XQD TXDOVLDVL SHUPXWD]LRQH HVVD SXz HVVHUH HVSUHVVD FRPH

σ = (m1 , m 2 )(m3 , m 4 ) GRYH mi ≠ m j SHU i ≠ j (i, j ) = 1..4 DEELDPR FLRq GHOOH WUDVSRVL]LRQL GLVJLXQWH VHJXH FKH LQ V4 • •

HVLVWH O·HOHPHQWR QHXWUR GHWHUPLQDWR GD σ e

HVLVWH OD SHUPXWD]LRQH LQYHUVD GL XQD TXDOVLDVL SHUPXWD]LRQH σ SRLFKp σ

•

−1

= (m3 , m 4 ) −1 (m1 , m 2 ) −1 = (m3 , m 4 )(m1 , m2 ) = (m1 , m 2 )(m3 , m 4 ) = σ

VL q DSSOLFDWD OD SURSULHWj FKH O·LQYHUVD GL XQD WUDVSRVL]LRQH q OD WUDVSRVL]LRQH VWHVVD H SHU L FLFOL GLVJLXQWL YDOH OD FRPPXWDWLYLWj SHU TXDQWR ULJXDUGD OD SURSULHWj GL FKLXVXUD VL RVVHUYL • σσ = ( m1 , m 2 )( m3 , m 4 )( m1 , m 2 )( m3 , m 4 ) = ( m1 , m 2 )( m3 , m4 )( m3 , m 4 )( m1 , m2 ) = σ e

e

σ

σe

•

PHQWUH SHU L SURGRWWL PLVWL HVHJXHQGR GLUHWWDPHQWH LO FDOFROR VL KD σ 18σ 20 = σ 20σ 18 = σ 22 σ 18σ 22 = σ 22σ 18 = σ 20 σ 20σ 22 = σ 22σ 20

= σ 18

V4 ULVXOWD SHUWDQWR XQ VRWWRJUXSSR GL A4 ,QROWUH QHOOD YHULILFD GHOOD SURSULHWj GL FKLXVXUD q HYLGHQ]LDWR DQFKH FKH L SURGRWWL VRQR FRPPXWDWLYL H TXLQGL ULVXOWD GLPRVWUDWD DQFKH O·DEHOLDQLWj GD FXL VHJXH OD QRUPDOLWj 6L RVVHUYL LQILQH FKH • V4 q XQ JUXSSR PDVVLPDOH GL A4 LQ TXDQWR A4 / V4 KD RUGLQH H TXLQGL HVVHQGR SULPR ULVXOWD VHPSOLFH FRQIURQWDUH LO FULWHULR GL PDVVLPDOLWj • JOL XQLFL VRWWRJUXSSL GL V4 VRQR {σ e , σ 18 } {σ e , σ 20 } {σ e , σ 22 } FKH ULVXOWDQR QRUPDOL LQ V4

PD

QRQ

LQ

A4

σ 20σ 18 (σ 20 ) −1 = σ 22σ 20 = σ 18

DG

HVHPSLR

SHU

LO

SULPR

σ 22σ 18 (σ 22 ) −1 = σ 20σ 22 = σ 18

GLPRVWUDWR QHO VHJXLWR 'LPRVWULDPR RUD FKH A4 QRQ DPPHWWH DOWUL VRWWRJUXSSL QRUPDOL SURSUL ROWUH V4 3DJ

VRWWRJUXSSR FRPH

YHUUj

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL 6XSSRQLDPR D WDOH VFRSR FKH N VLD XQ VRWWRJUXSSR GL A4 GLVWLQWR GD V4 6L SRVVRQR DOORUD YHULILFDUH GXH FDVL • N ⊂ V4 •

N ⊄ V4 H TXLQGL N FRQWLHQH DOPHQR XQ HOHPHQWR DSSDUWHQHQWH D V

&DVR N ⊂ V 4 ,Q TXHVWR FDVR N GHYH HVVHUH XQ VRWWRJUXSSR GL V4 H TXLQGL QRQ SXz FRQWHQHUH WUH SHUPXWD]LRQL LQ TXDQWR VH (σ i , σ j ) ∈ V4 FRQ σ i

≠ σ j H (σ i ≠ σ e , σ j ≠ σ e ) σ iσ j = σ k ≠ σ e FRQ

(k ≠ j, k ≠ i ) $OORUD N SXz HVVHUH VROR XQR GHL VHJXHQWL VRWWRJUXSSL GL V4 {σ e , σ 18 } {σ e , σ 20 } {σ e , σ 22 } 7DOL VRWWRJUXSSL VRQR QRUPDOL LQ V4 PD QRQ VRQR QRUPDOL LQ A4 FRPH VL SXz GLPRVWUDUH IDFLOPHQWH FRQ L VHJXHQWL FRQWURHVHPSL •

σ 6σ 18 (σ 6 ) −1 = (13)( 12)(12)(34)(13) −1 (12) −1 = (13)(34)(13)(12) = (142) = σ 10 ∉ {σ e , σ 18 }

σe

H TXLQGL {σ e , σ 18 } QRQ q QRUPDOH A4 •

σ 6σ 20 (σ 6 ) −1 = (13)(12)(13)(24)(13) −1 (12) −1 = (13)(12)(13)(24)(13)(12) = = (13)(12)(13)(13)(24)(12) = (13)(12)(24)(12) = (134) = σ 8 ∉ {σ e , σ 20 }

σe

H TXLQGL {σ e , σ 20 } QRQ q QRUPDOH LQ A4 •

σ 16σ 22 (σ 16 ) −1 = (24)(23)(14)(23)(24) −1 (23) −1 = (24)(23)(14)(23)(24)(23) = = (24)(23)(23)(14)(24)(23) = (24)(14)(24)(23) = (132) = σ 6 ∉ {σ e , σ 22 }

σe

H TXLQGL {σ e , σ 22 } QRQ q QRUPDOH LQ A4 3RVVLDPR TXLQGL FRQFOXGHUH FKH QRQ HVLVWRQR VRWWRJUXSSL QRUPDOL DO JUXSSR DOWHUQR A4 FRQWHQXWL LQ V4 &DVR N ⊄ V 4

,Q TXHVWR FDVR FRPH JLj RVVHUYDWR LQ SUHFHGHQ]D HVLVWH DOPHQR XQD SHUPXWD]LRQH σ i ∈ N WDOH

σ i ∈ V 6L VXSSRQJD FKH VLD N A4 H VL VXSSRQJD FKH σ i = σ 2 = (123)

6LD LQROWUH ρ ∈ V XQD TXDOVLDVL DOWUD SHUPXWD]LRQH GL V HVSULPLELOH TXLQGL FRPH VLQJROR FLFOR VRWWR WDOL LSRWHVL

σ = ρ −1σ 2 ρ ∈ N SRLFKp N A4 ,QROWUH VL RVVHUYL FKH ρ H ρ

−1

SRVVRQR HVVHUH HVSUHVVH LQ IRUPD HVWHVD FRPH VHJXH

2 3 4 · −1 §1ρ 2 ρ 3ρ 4 ρ · §1 ¸¸ ρ = ¨¨ ¸ ρ = ¨¨ 2 3 4 ¸¹ ©1ρ 2 ρ 3ρ 4 ρ ¹ ©1

$OORUD OD SHUPXWD]LRQH σ YDOH

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL

2 3 4 · §1ρ 2 ρ 3ρ 4 ρ · §1 ¸¸(123)¨¨ ¸¸ = (1ρ 2 ρ 3ρ ) σ = ¨¨ 2 3 4 ¹ ©1 ©1ρ 2 ρ 3ρ 4 ρ ¹

&DOFROLDPR RUD LO YDORUH GL σ SHU RJQL ρ ∈ V ρ = (123) ρ = (124) ρ = (132) ρ = (134) ρ

= (142) ρ = (143) ρ = (234) ρ = (243) σ = (123) σ = (124) σ = (132) σ = (134) σ = (142) σ = (143) σ = (234) σ = (243)

5LVXOWD HYLGHQWH FKH σ DVVXPH TXDOVLDVL SHUPXWD]LRQH GL V RVVLD LQ DOWUL WHUPLQL WXWWL L ²FLFOL GL V VRQR WUD ORUR FRQLXJDWL 4XLQGL VLFFRPH σ ∈ N FLz LPSOLFD FKH N ⊃ V RVVLD FKH N FRQWLHQH WXWWL L FLFOL GL V GD FXL VHJXH N ≡ A4 6L VXSSRQJD RUD FKH DG N DSSDUWHQJD QRQ OD VSHFLILFD SHUPXWD]LRQH (123) PD XQD JHQHULFD SHUPXWD]LRQH σ i ∈ V 2UD VLFFRPH σ i H

(123) VRQR FRQLXJDWH RVVLD DEELDPR YHULILFDWR GLUHWWDPHQWH FKH HVLVWH XQD SHUPXWD]LRQH ρ ∈ A4 WDOH FKH

σ i = ρ −1 (123) ρ ∈ N VL GHGXFH

(123) = ρσ i ρ −1 ∈ N SRLFKp N A4 ,Q TXHVWR PRGR FL VLDPR ULFRQGRWWL DO FDVR SUHFHGHQWH LQ FXL (123) ∈ N 3HUWDQWR DQFKH QHO FDVR JHQHUDOH VL KD N ≡ A4 3RVVLDPR GXQTXH ULHSLORJDUH FKH VH N A4 N ≡ A4 H TXLQGL N QRQ SXz HVVHUH XQ VRWWRJUXSSR SURSULR GL A4 LO TXDOH SHUWDQWR DPPHWWH FRPH VRWWRJUXSSR SURSULR QRUPDOH VROR V4 2VVHUYD]LRQH 1HOOD GLPRVWUD]LRQH DEELDPR DYXWR FRPH ULVXOWDWR FKH WXWWL L FLFOL GL A4 ULVXOWDQR FRQLXJDWL LQ PRGR DQDORJD D TXDQWR DYYLHQH SHU An FRQ n ≥ 5

6RWWRJUXSSL QRUPDOL GL Σ 4

,Q PRGR VLQWHWLFR LO JUXSSR Σ 4 SXz HVVHUH HVSUHVVR FRPH GL VHJXLWR ULSRUWDWR Σ 4 = {D4 , A4 } = {D4 , V , V4 } 6L YXROH GLPRVWUDUH FKH Σ 4 DPPHWWH FRPH VRWWRJUXSSR QRUPDOH SURSULR VROR LO JUXSSR DOWHUQR A4 HG LO JUXSSR V4 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL SDVVR GLPRVWULDPR FKH A4 Σ 4 $EELDPR JLj GLPRVWUDWR OD QRUPDOLWj GHO JUXSSR DOWHUQR ULVSHWWR DO JUXSSR VLPPHWULFR QHO FDVR JHQHUDOH SHU RJQL YDORUH GHOO·RUGLQH GHO JUXSSR VLPPHWULFR VWHVVR SDVVR GLPRVWULDPR FKH V4 Σ 4 1HOOR VWXGLR GHOOD VWUXWWXUD GL V4 DEELDPR YHULILFDWR FKH V4 q XQ JUXSSR DEHOLDQR H TXLQGL ULVXOWD QHFHVVDULDPHQWH QRUPDOH LQ Σ 4 SDVVR GLPRVWULDPR FKH QRQ HVLVWH XQ DOWUR VRWWRJUXSSR QRUPDOH /D VWUDWHJLD GHOOD GLPRVWUD]LRQH q TXHOOD GL YHULILFDUH FKH TXDOVLDVL VRWWRJUXSSR QRUPDOH GL Σ 4 QRQ FRQWHQXWR LQ A4 FRLQFLGH FRQ Σ 4 6LD GXQTXH N Σ 4 WDOH FKH N ⊄ A4 SHUWDQWR O·LQVLHPH HOHPHQWR

N * = N ∩ A4 GHYH DYHUH DOPHQR XQ

N * FRPH LQWHUVH]LRQH GL GXH VRWWRJUXSSL QRUPDOL ULVXOWD VRWWRJUXSSR QRUPDOH VLD LQ N VLD LQ A4 H TXLQGL

N * ≡ V4 LQ TXDQWR DEELDPR GLPRVWUDWR FKH O·XQLFR VRWWRJUXSSR QRUPDOH GL A4 q V4 'RYHQGR HVVHUH N ⊄ A4 H N

*

= N ∩ A4 = V4 VHJXH N ⊂ V4

H FRQWLHQH DOPHQR XQD SHUPXWD]LRQH ρ ∈ D 4

2UD ρ ∈ D 4 SXz HVVHUH XQD WUDVSRVL]LRQH RSSXUH XQ FLFOR GL RUGLQH LQ TXHVWR XOWLPR FDVR VL KD ρ = (m1 , m 2 , m3 , m4 ) FRQ mi ∈ {1,2,3,4} SHU i = 1..4 6H PROWLSOLFR LO VXGGHWWR FLFOR SHU XQD TXDOVLDVL SHUPXWD]LRQH GL σ = ( m1 , m2 )(m3 , m 4 ) ∈ V4 VL RWWLHQH

σρ = (m1 , m2 )(m3 , m4 )(m1 , m2 , m3 , m 4 ) = (m1 , m3 ) ∈ N

4XLQGL DQFKH QHO FDVR LQ FXL ρ IRVVH XQ FLFOR N FRQWLHQH DOPHQR XQD WUDVSRVL]LRQH H SHUWDQWR LQ RJQL FDVR N FRQWLHQH DOPHQR XQD WUDVSRVL]LRQH 6XSSRQLDPR FKH WDOH SHUPXWD]LRQH DSSDUWHQHQWH D N VLD GDWD GD (12) GHWWD LQROWUH σ ∈ Σ 4 XQD TXDOVLDVL WUDVSRVL]LRQH GHO JUXSSR VLPPHWULFR VL KD

σ −1 (12)σ ∈ N LQ TXDQWR N Σ 4 HG LQROWUH LQ PRGR DQDORJR D TXDQWR YLVWR LQ SUHFHGHQ]D VL SXz SRUUH

σ −1 (12)σ = (1σ 2σ ) = ε ∈ N

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL 'D FXL VHJXH FKH RJQL WUDVSRVL]LRQH σ ∈ Σ 4 DSSDUWLHQH DQFKH D N FRPH ULVXOWD GDOOD YDOXWD]LRQH GL σ

−1

(12)σ = ε DO YDULDUH GL σ

σ = (12) σ = (13) σ = (14) σ = (23) σ = (24) σ = (34) ε = (12) ε = (13) ε = (14) ε = (23) ε = ( 24) ε = (34)

2UD VH OD WUDVSRVL]LRQH DSSDUWHQHQWH D N LQYHFH GL HVVHUH OD

(12) IRVVH XQD TXDOVLDVL DOWUD

WUDVSRVL]LRQH (m1 , m 2 ) VLFFRPH WXWWH OH WUDVSRVL]LRQL VRQR FRQLXJDWH HVLVWH XQD WUDVSRVL]LRQH

σ ∈ Σ 4 WDOH FKH

σ −1 (m1 , m2 )σ = (m1σ m2σm2 ) = (12) ∈ N H SHUWDQWR FL VLDPR ULFRQGRWWL DO FDVR LQ FXL N FRQWLHQH (12) H TXLQGL SRVVLDPR DIIHUPDUH FKH LQ RJQL FDVR N FRQWLHQH WXWWH OH WUDVSRVL]LRQL GL Σ 4 GD FXL VHJXH N ≡ Σ 4 H FLz FRPSOHWD OD GLPRVWUD]LRQH

6WUXWWXUD GHu JUXSSL TXR]LHQWH Σ 4 / A4 H A 4 /V4

3HU TXDQWR ULJXDUGD LO JUXSSR Σ 4 / A4 YDOJRQR OH VHJXHQWL SURSULHWj • KD GXH VROL HOHPHQWL Σ 4 / A4 = {A4 , D4 } •

q SULPR H TXLQGL ULVXOWD DQFKH FLFOLFR HG DEHOLDQR FRQ HOHPHQWR JHQHUDWRUH D4

( D4 ) 2 = A4 • q VHPSOLFH 3HU TXDQWR ULJXDUGD LO JUXSSR A 4 /V 4 YDOJRQR OH VHJXHQWL SURSULHWj

•

KD RUGLQH SDUL D Ord ( A 4 ) / Ord (V4 ) = 12 / 4 = 3 SHUWDQWR KD WUH HOHPHQWL

•

q SULPR H TXLQGL ULVXOWD FLFOLFR HG DEHOLDQR

5LVROXELOLWj GHO JUXSSR Σ 4

'D WXWWR TXDQWR SUHFHGH SRVVLDPR GHGXUUH FKH LO JUXSSR VLPPHWULFR GL RUGLQH ULVXOWD ULVROXELOH LQ TXDQWR • (VLVWH OD FDWHQD GL VRWWRJUXSSL Σ 4 ⊃ A4 ⊃ V4 ⊃ {e} • •

A4 Σ 4 V4 A4 {e} V4 Σ 4 / A4 A4 /V4 V4 /{e} = V4 VRQR DEHOLDQL 2VVHUYD]LRQH

/D ULVROXELOWj GHO JUXSSR Σ 4 q FRUUHODWD DOO·HVLVWHQ]D GL XQD IRUPXOD ULVROXWLYD SHU UDGLFDOL GHOOH HTXD]LRQL DOJHEULFKH GL TXDUWR JUDGR

*UXSSL VLPPHWULFR GL RUGLQH PDJJLRUH GL

1HO SUHVHQWH SDUDJUDIR DQDOL]]LDPR OD VWUXWWXUD GHL JUXSSL VLPPHWULFL Σ n FRQ n ≥ 5 $ WLWROR GL HVHPSLR VL ULSRUWDQR OH WDEHOOH FRQ O·LQVLHPH GHOOH SHUPXWD]LRQL GL Σ 5 H GL A5 LQ RJQL FRORQQD FL VRQR ULSRUWDWH OH SHUPXWD]LRQL ULVSHWWR DOOD VHTXHQ]D 12355

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL 3HUPXWD]LRQL GHO *UXSSR VLPPHWULFR GL 2UGLQH

3HUPXWD]LRQL GHO *UXSSR $OWHUQR GL 2UGLQH

6HPSOLFLWj GHO JUXSSR DOWHUQR A5

,O JUXSSR DOWHUQR A5 q VHPSOLFH D GLIIHUHQ]D GL A4 'LPRVWULDPR TXDQWR VRSUD HQXQFLDWR /D VWUDWHJLD GHOOD GLPRVWUD]LRQH q TXHOOD GL VXSSRUUH FKH HVLVWH XQ VRWWRJUXSSR QRUPDOH N GHO JUXSSR DOWHUQR A5 H GL YHULILFDUH FKH QHFHVVDULDPHQWH N ≡ A5 H FKH TXLQGL QRQ HVLVWRQR VRWWRJUXSSL QRUPDOL SURSUL GL A5 3HU FRQVHJXLUH WDOH ULVXOWDWR DEELDPR GXH SURSULHWj LPSRUWDQWL GD SRWHU XWLOL]]DUH • N GHYH FRQWHQHUH DOPHQR XQD SHUPXWD]LRQH GLYHUVD GDOO·LGHQWLWj FKH ODVFLD ILVVR XQ HOHPHQWR GL {1,2,3,4,5} • VH N FRQWLHQH XQ FLFOR DOORUD FRLQFLGH FRQ A5 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL 3HUWDQWR VL GHYH FHUFDUH GL XWLOL]]DUH OD SULPD SURSULHWj SHU GLPRVWUDUH FKH DOPHQR XQ FLFOR q FRQWHQXWR LQ N SRLFKp GD FLz VL GHGXFH FKH N ≡ A5 6LD GXQTXH N XQ VRWWRJUXSSR SURSULR GL A5 WDOH FKH N A5 H VLD H = {σ ∈ A5 : 5σ = 5}

QDWXUDOPHQWH H ≈ A4 3HU OH SURSULHWj GHOOD LQWHUVH]LRQH GL JUXSSL QRUPDOL VL KD

N* = N ∩ H H 3RLFKp N q XQ VRWWRJUXSSR QRUPDOH GL A5 GHYH FRQWHQHUH DOPHQR XQD SHUPXWD]LRQH ρ GLYHUVD GDOO·LGHQWLWj FKH ODVFLD ILVVR XQ HOHPHQWR a ∈ {1,2,3,4,5} SHUWDQWR YDOH aρ = a 6L VFHOJD RUD XQD SHUPXWD]LRQH σ ∈ A5 WDOH FKH aσ = 5 VL KD •

σ * = σ −1 ρσ ∈ N LQ TXDQWR N A5

•

5σ * = 5(σ −1 ρσ ) = 5(σ −1 )( ρσ ) = a ( ρσ ) = (, aρ )σ = aσ = 5 σ * ∈ H a

a

$OORUD σ

*

= σ −1 ρσ ∈ N * LQ TXDQWR DSSDUWLHQH FRQWHPSRUDQHDPHQWH D N HG D H H GXQTXH

N * ULVXOWD XQ VRWWRJUXSSR QRUPDOH SURSULR GL H LQ TXDQWR FRQWLHQH DOPHQR XQ HOHPHQWR

GLYHUVR GDOO·XQLWj 2UD DEELDPR QRWDWR FKH H ≈ A4 RVVLD H KD OH VWHVVH SHUPXWD]LRQL GL A4 LQ FXL VHPSOLFHPHQWH q ODVFLDWR ILVVR O·HOHPHQWR 5 'D FLz VL GHGXFH FKH SRLFKp A4 DPPHWWH FRPH VRWWRJUXSSR QRUPDOH SURSULR VROR V4 GHYH HVVHUH

N * ≡ A4 RSSXUH N * ≡ V4 &DVR N

*

≡ A4

,Q TXHVWR FDVR N FRQWLHQH WXWWL L ²FLFOL GL A4 SRLFKp

A4 ≡ N * = N ∩ H = N ∩ A4 H TXLQGL

N ≡ A5 H OD GLPRVWUD]LRQH q FRQFOXVD *

≡ V4 ,Q TXHVWR FDVR N FRQWLHQH WXWWH OH SHUPXWD]LRQL GL V4 FKH VRQR GHOOD IRUPD ( m1 m2 )(m3 m4 ) GRYH JOL HOHPHQWL VRQR WXWWL GLVWLQWL HG DSSDUWHQJRQR DOO·LQVLHPH {1,2,3,4} &DVR N

6LD ρ ∈ A5 VWDQWH OD QRUPDOLWj GL N VL KD FKH ρ

−1

(m1m 2 )(m3 m4 ) ρ ∈ N

,QROWUH HVVHQGR L GXH FLFOL GLVJLXQWL YDOH

ρ −1 (m1m2 )(m3 m4 ) ρ = (m1 ρ m2 ρ ) (m3 ρ m4 ρ )

3RQHQGR QHOOD UHOD]LRQH SUHFHGHQWH • ( m1 m2 )( m3 m 4 ) = (13)( 24) • ρ = ( 245) = ( 24)(45) ∈ A5 VL RWWLHQH

(245) −1 (13)(24)(245) = (13)(45) ∈ N 'D FXL VHJXH

(13)(24)(13)(45) = (13)(13)(24)(45) = (245) ∈ N

σe

3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL 3HUWDQWR VL SXz FRQFOXGHUH OD GLPRVWUD]LRQH RVVHUYDQGR FKH SRLFKp N FRQWLHQH LO FLFOR (245) VHJXH FKH N ≡ A5

6HPSOLFLWj GHO JUXSSR DOWHUQR An

,O JUXSSR DOWHUQR An q VHPSOLFH SHU n ≥ 5 $EELDPR JLj GLPRVWUDWR LO FDVR n = 5 3HU n > 5 XWLOL]]LDPR LO PHWRGR GL LQGX]LRQH /·LSRWHVL LQGXWWLYD LQL]LDOH H FKH LO JUXSSR DOWHUQR A5 q VHPSOLFH FRPH GLPRVWUDWR QHO SDUDJUDIR SUFHGHQWH 3RQLDPR SHU LQGX]LRQH FKH An −1 GHGXFHQGR FKH GD FLz VHJXH OD VHPSOLFLWj GL An /D VWUDWHJLD GHOOD GLPRVWUD]LRQH q DQDORJD D TXHOOD XWLOL]]DWD QHO FDVR GL A5

6LD GXQTXH N XQ VRWWRJUXSSR SURSULR GL An WDOH FKH N An H VLD H = {σ ∈ An : nσ = n}

'DOOD GHILQL]LRQH GL H VHJXH H ≈ An −1 3HU OH SURSULHWj GHOOD LQWHUVH]LRQH GL JUXSSL QRUPDOL VL KD

N* = N ∩ H H 3RLFKp N q XQ VRWWRJUXSSR QRUPDOH GL An GHYH FRQWHQHUH DOPHQR XQD SHUPXWD]LRQH ρ GLYHUVD GDOO·LGHQWLWj FKH ODVFLD ILVVR XQ HOHPHQWR a ∈ {1,2,3,4,5,...n} SHUWDQWR YDOH aρ

= a

6L VFHOJD RUD XQD SHUPXWD]LRQH σ ∈ An WDOH FKH aσ = n VL KD •

σ * = σ −1 ρσ ∈ N LQ TXDQWR N An

•

nσ * = n(σ −1 ρσ ) = n(σ −1 )( ρσ ) = a ( ρσ ) = (, aρ )σ = aσ = n σ * ∈ H a

a

$OORUD σ

*

= σ −1 ρσ ∈ N * LQ TXDQWR DSSDUWLHQH FRQWHPSRUDQHDPHQWH D N HG D H H GXQTXH

N * ULVXOWD XQ VRWWRJUXSSR QRUPDOH SURSULR GL H LQ TXDQWR FRQWLHQH DOPHQR XQ HOHPHQWR

GLYHUVR GDOO·XQLWj 2UD DEELDPR QRWDWR FKH H ≈ An −1 RVVLD

H KD OH VWHVVH SHUPXWD]LRQL GL An −1 LQ FXL

VHPSOLFHPHQWH q ODVFLDWR ILVVR O·HOHPHQWR n 'D FLz VL GHGXFH FKH

N * = An −1 LQ TXDQWR

An −1

q VHPSOLFH H TXLQGL DPPHWWH FRPH VRWWRJUXSSL QRUPDOL VROR VH VWHVVR H LO JUXSSR {e} FRQWHQHQWH VROR OD SHUPXWD]LRQH XQLWj • H ≠ {e} SRLFKp DEELDPR YLVWR FKH FRQWLHQH DOPHQR XQD SHUPXWD]LRQH GLYHUVD GDOOD SHUPXWD]LRQH XQLWj

•

3RVVLDPR TXLQGL FRQFOXGHUH FKH N ≡ An SRLFKp N

*

≡ An −1 ⊂ N H TXLQGL N FRQWLHQH WXWWL L FLFOL

GL An −1

6RWWRJUXSSL QRUPDOL GHO JUXSSR VLPPHWULFR Σ n

,O JUXSSR VLPPHWULFR Σ n QRQ DPPHWWH DOWUL VRWWRJUXSSL QRUPDOL QRQ EDQDOL D GLIIHUHQ]D GHO JUXSSR DOWHUQR An 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL 'LPRVWULDPR TXDQWR VRSUD HQXQFLDWR 6LD N XQ VRWWRJUXSSR SURSULR GL Σ n WDOH FKH •

N Σ n

•

N QRQ VLD FRQWHQXWR LQ An RVVLD N ⊄ An

6L FRQVLGHUL O·LQVLHPH N ∩ An FKH ULVXOWD HVVHUH XQ VRWWRJUXSSR QRUPDOH GL An VL YHGDQR OH SURSULHWj GHOO·LQWHUVH]LRQH GL JUXSSL QRUPDOL 3RLFKp An q VHPSOLFH H TXLQGL QRQ DPPHWWH VRWWRJUXSSL QRUPDOL SURSUL VL SRVVRQR YHULILFDUH GXH FDVL &DVR N ∩ An = An 'DOO·DSSOLFD]LRQH GHO OHPPD LOOXVWUDWR QHO SDUDJUDIR UHODWLYR DL JUXSSL DOWHUQL HVVHQGR N QRQ VWUHWWDPHQWH FRQWHQXWR LQ An VHJXH Ord ( N )

= 2Ord ( N ∩ An ) = 2Ord ( An ) = 2

n! = n! 2

$OORUD SRLFKp SHU LSRWHVL N Σ n H G DYHQGR N OR VWHVVR QXPHUR GL HOHPHQWL GL Σ n VHJXH

N ≡ Σ n &DVR N ∩ An = {e} 6H $SSOLFDQGR VHPSUH LO OHPPD VRSUD ULFKLDPDWR QRQ YHULILFDQGRVL SHU LSRWHVL LO FDVR N ⊂ An VHJXH Ord ( N ) = 2Ord ( N ∩ An ) = 2Ord ({e}) = 2

$OORUD VL GHGXFH FKH N GHYH HVVHUH XQ VRWWRJUXSSR GL RUGLQH GXH GHO JUXSSR Σ n H GHYH LQROWUH HVVHUH FRVWLWXLWR GDOOH VROH SHUPXWD]LRQL GLVSDUL HVVHQGR N ∩ An = {e}

0D WDOH FRQFOXVLRQH q DVVXUGD LQ TXDQWR LO JUXSSR VLPPHWULFR Σ n FRQ n > 2 QRQ DPPHWWH VRWWRJUXSSL QRUPDOL GL RUGLQH SHU n = 2 A2 ≡ {e} H N ≡ Σ 2 HVVHQGR XQ VRWWRJUXSSR GL RUGLUQH 4XDQWR VRSUD DIIHUPDWR VL SXz GLPRVWUDUH ULFRUGDQGR FKH • RJQL SHUPXWD]LRQH VL SXz IDWWRUL]]DUH FRPH SURGRWWR GL FLFOL GLVJLXQWL • SUHVD XQD TXDOVLDVL SHUPXWD]LRQH ρ ∈ Σ n HG XQ TXDOVLDVL FLFOR ( m1 , m 2 ,.....m k ) GL RUGLQH

k ≤ n YDOH OD VHJXHQWH UHOD]LRQH ρ −1 (m1 , m2 ,.....m k ) ρ = (m1 ρ , m 2 ρ ,.....m k ρ ) $OORUD VH N = {σ e , σ } Σ n QHFHVVDULDPHQWH σ = σ 1σ 2 ....σ h GRYH RJQL σ i q XQ FLFOR GHO WLSR

σ i = (m1,σ i , m2,σ i ,.....mk ,σ i ) GLVJLXQWR FRQ JOL DOWUL H SHUWDQWR

ρ −1σρ = ρ −1σ 1σ 2 ....σ h ρ = ρ −1σ 2 ρρ −1σ 3 ρρ −1σ 4 ρ ..........ρ −1σ k ρ GRYH

ρ −1σ i ρ = (m1,σ i ρ , m2,σ i ρ ,.....mk ,σ i ρ ) SHU i = 1..k 2UD VH N = {σ e , σ } Σ n VHJXH

−1

ρ σρ ∈ N GD FXL R ρ −1σρ = σ R ρ −1σρ = σ e PD FLz q LPSRVVLELOH SHU RJQL ρ ∈ Σ n LQ TXDQWR ρ SHUPXWD L VLQJROL HOHPHQWL GL σ 3RVVLDPR TXLQGL FRQFOXGHUH FKH Σ n FRQ n > 2 QRQ DPPHWWH VRWWRJUXSSL QRUPDOL GL RUGLQH H TXLQGL LO FDVR FKH VWLDPR DQDOL]]DQGR QRQ SXz YHULILFDUVL 3DJ

)RQGDPHQWL GL $OJHEUD &DSLWROR ² *UXSSL 6LPPHWULFL &RQ FLz VL SXz FRQFOXGHUH FKH ULPDQH SRVVLELOH VROR LO FDVR GD FXL VHJXH FKH VH N Σ n FRQ

N ⊄ An QHFHVVDULDPHQWH N ≡ Σ n H TXLQGL Σ n QRQ KD DOWUL VRWWRJUXSSL QRUPDOL ROWUH An

1RQ ULVROXELOLj GHO JUXSSR Σ n

,O JUXSSR VLPPHWULFR Σ n GL RUGLQH n ≥ 5 QRQ q ULVROXELOH ,QIDWWL DEELDPR YLVWR FKH Σ n SHU n ≥ 5 DPPHWWH FRPH VRWWRJUXSSL QRUPDOL SURSUL VROR LO JUXSSR DOWHUQR An LO TXDOH ULVXOWD VHPSOLFH 3HUWDQWR SXz HVLVWHUH VROR OD VHJXHQWH FDWHQD GL VRWWRJUXSSL QRUPDOL Σ n ⊃ An ⊃ {e} An Σ n {e} An

Σ n SXz DOORUD HVVHUH ULVROXELOH VROR VH Σ n / An H An /{e} = An VRQR DEHOLDQL

2UD Σ n / An q DEHOLDQR LQ TXDQWR JUXSSR SULPR

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GHQRPLQDWRUH GHOOH HVSUHVVLRQL FKH IRUQLVFRQR LO YDORUH GHOOD WHUQD GL LQFRJQLWH ( x

½ a11 a12 a13 ° Det ( A) ¾ = a21 a22 a23 = Det (aij )°¿ a31 a32 a33 = a11a22 a33 − a11a23a32 − a12 a21a33 + a12 a23a31 + a13a21a32 − a13a22 a31 A

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A = a11a22a33 − a11a23a32 − a12a21a33 + a12a23a31 + a13a21a32 − a13a22a31 = = a11(a22a33 − a23a32 ) − a12 (a21a33 − a23a31) + a13 (a21a32 − a22a31) 6L RVVHUYL RUD FKH

•

a22 a32

a23 = a22 a33 − a 23 a32 a33

•

a21 a31

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a21 a31

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•

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3DJ

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a11 A = a21 a31

a12 a22 a32

a13 a22 a23 = a11 a32 a33

a23 a21 − a12 a33 a31

a23 a21 + a13 a33 a31

a22 a32

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a12

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(i+ j )

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a21 a31

j ) q GLVSDUL

a23 a + (−1) (1+ 2) a12 21 a33 a31

a23 + a33

a22 a32

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a11 WDEHOOD a21 a31

a12 a22 a32

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a11 a21 a31

a12 a22 a32

a13 a22 a23 a32 a33

a23 a33

•

a11 a21 a31

a12 a 22 a32

a13 a 21 a23 a31 a33

a23 a33

•

a11 a21 a31

a12 a 22 a32

a13 a 21 a23 a31 a33

a22 a32

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A = a11a22a33 − a11a23a32 − a12a21a33 + a12a23a31 + a13a21a32 − a13a22a31 = = a11(a22a33 − a23a32 ) − a21 (a12a33 − a13a32 ) + a31 (a12a23 − a22a13 ) 3DJ

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•

a22 a32

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•

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•

a12 a13

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a11 A = a21 a31

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a23 a12 − a21 a33 a23

a13 a12 + a31 a33 a13

a22 a23

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a11

a12

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a13

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a12 a13

a23 a + (−1) (1+ 2) a21 12 a33 a23

a13 + a33

a22 a23

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•

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a11 a21 a31

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a11 a21 a31

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a11

a12

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a23 a + (−1) ( 2+ 2) a22 11 a33 a31

a13 + a33

a11 a13 a21 a23

6L RWWLHQH GXQTXH

a11 A = a21 a31

a12 a22 a32

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a23 a11 + a22 a33 a31

a13 a11 − a32 a33 a21

a13 a23

9HGLDPR XQ HVHPSLR QXPHULFR

1 2 0 3 1 3 = (−1) (1+1) 1 2 0 1

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= 1 − 2(3 − 6) + 1(−2) = 5

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ª­ε 111 = ε 112 = ε 113 = ε 121 = ε 122 = ε 131 = ε 133 = 0 º «°° 211 » = ε 212 = ε 221 = ε 222 = ε 223 = ε 232 = ε 233 = 0» § i = 1..3 · «®ε «° 311 Ȭ ¸ 313 = ε 322 = ε 323 = ε 331 = ε 332 = ε 333 = 0 » ¨ j = 1..3 ¸ = «¯°ε = ε « 123 » ¨ k = 1..3 ¸ 231 ¹ = ε 312 = +1 «­°ε = ε »© «® 132 » 213 = ε 321 = −1 ¬«°¯ε = ε ¼»

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SHUPXWD]LRQH SDUL HVHPSLR

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•

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•

jik

= −1

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( jik ) q GLVSDUL

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ε

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p

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qrh

= (−1) p ε 123 ε qrh = (−1) p ε qrh = ε qrhε ijk

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A

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a12 a22 a32

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a11 a21 a31

a12 a22 a32

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A = a11a22a33 − a11a32a23 − a21a12a33 + a31a12a23 + a21a32a13 − a31a22a13 RVVLD FRPH VRPPD GL WHUPLQL a1i a2 j a3k LQ FXL 3DJ

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4XHVWR GLVFRUVR SXz HVVHUH XOWHULRUPHQWH JHQHUDOL]]DWR YHGLDPR DOFXQL HVHPSL

A = ε kij a1k a2i a3 j = ε kij a2i a1k a3 j 6L ULFRUGL RUD FKH ε kij = ε 312ε ijk = ε 231ε ijk SHUWDQWR VL SXz

SRUUH

(

ε 231 A = ε ijk a2i a3 j a1k

(

)

A = ε kij a1k a2i a3 j = ε kij a2i a3 j a1k = ε 231 ε ijk a2i a3 j a1k GD FXL VHJXH

) ,O YDORUH ε

231 SXz HVVHUH SRUWDWR D SULPR PHPEUR HVVHQGR SDUL

A =ε

jik

a1 j a2i a3k = ε

jik

a2i a1 j a3k 6L ULFRUGL RUD FKH ε

A =ε

jik

a1 j a2i a3k = ε

jik

a2i a1 j a3k = ε 213 ε ijk a2i a1 j a3k GD

(

ε 213 A = ε ijk a2i a1 j a3k

(

jik

)

= ε 213ε ijk SHUWDQWR VL SXz SRUUH FXL

VHJXH

) ,O YDORUH ε 213 HVVHQGR XJXDOH D SXz HVVHUH SRUWDWR D SULPR

PHPEUR PROWLSOLFDQGR HQWUDPEL L PHPEUL SHU 'DL SUHFHGHQWL GXH HVHPSL SRVVLDPR GHGXUUH O·HVSUHVVLRQH SL JHQHUDOH GHOOR VYLOXSSR GL XQ GHWHUPLQDQWH ,QIDWWL GHWWD pqr XQD SHUPXWD]LRQH GHOOD WHUQD ULVXOWD

(

ε pqr A = ε ijk a pi aqj ark

)

3URSULHWj GHL GHWHUPLQDQWL GHO WHU]R RUGLQH

8Q GHWHUPLQDQWH GHO WHU]R RUGLQH q VYLOXSSDELOH FRPH VRPPD GL GHWHUPLQDQWL GL RUGLQH GXH GD TXHVWD RVVHUYD]LRQH q IDFLOPHQWH GHGXFLELOH OD YDOLGLWj SHU L GHWHUPLQDQWL GL RUGLQH WUH GL WXWWH OH SURSULHWj GLPRVWUDWH QHO FDVR GL RUGLQH GXH 3HU FRPSOHWH]]D YHQJRQR ULSRUWDWH OH GLPRVWUD]LRQL IDFHQGR XVR GHOO·HVSUHVVLRQH GHO GHWHUPLQDQWH WUDPLWH O·LQGLFDWRUH GL /HYL &LYLWD

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A XQD PDWULFH H AT OD VXD WUDVSRVWD VL KD

aij T = a ji 'D FLz VHJXH

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'HWHUPLQDQWH GL XQD PDWULFH FRQ ULJKH R FRORQQH VFDPELDWH ULVSHWWR DG XQD PDWULFH GDWD

)LVVDWD XQD PDWULFH FKH •

A VLD B OD PDWULFH RWWHQXWD GD A VFDPELDQGR GL SRVWR DOOH ULJKH LQ PRGR WDOH

b1i = aqi (i = 1,2,3) RVVLD JOL HOHPHQWL GHOOD SULPD ULJD GL B VRQR GDWL GDJOL HOHPHQWL GHOOD

q _ esima ULJD GL A •

b2i = ari (i = 1,2,3) RVVLD JOL HOHPHQWL GHOOD VHFRQGD ULJD GL B VRQR GDWL GDJOL HOHPHQWL GHOOD

r _ esima ULJD GL A •

b3i = asi (i = 1,2,3) RVVLD JOL HOHPHQWL GHOOD WHU]D ULJD GL B VRQR GDWL GDJOL HOHPHQWL GHOOD s _ esima ULJD GL A

5LFRUGDQGR O·HVSUHVVLRQH JHQHUDOH GHO GHWHUPLQDQWH VL KD

(

)

B = ε ijk b1i b2 j b3k = ε ijk aqi arj a sk = ε qrs A 3DJ

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A FDPELDWR GL VHJQR − A VH OD SHUPXWD]LRQH (qrs ) q XQD SHUPXWD]LRQH GLVSDUL ULVSHWWR DOOD SHUPXWD]LRQH IRQGDPHQWDOH (123)

GHWHUPLQDQWH GL

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RWWHQXWD

§ a 21 ¨ B = ¨ a 22 ¨a © 23 GD

A

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OD

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a11 a12 a13

a31 · ¸ a32 ¸ a33 ¸¹

FRORQQD

FRQ

OD

VHFRQGD

QHO

TXDO

FDVR

(qrs ) = (213) ε qrs = ε 213 = −1 B = − A ,Q VRVWDQ]D VH VL HVHJXH XQ QXPHUR SDUL GL VFDPEL VXOOH ULJKH R VXOOH FRORQQH LO VHJQR GHO GHWHUPLQDQWH QRQ FDPELD PHQWUH VL KD XQ LQYHUVLRQH GL VHJQR QHO FDVR GL XQ QXPHUR GLVSDUL GL VFDPEL

'HWHUPLQDQWH GL XQD PDWULFH FRQ XQD ULJD FRORQQD PROWLSOLFDWD SHU XQD FRVWDQWH

6LD A′ XQ D PDWULFH RWWHQXWD GD SULPD ULJD VL KD

A PROWLSOLFDQGR XQD VXD ULJD SHU XQD FRVWDQWH λ DG HVHPSLR OD

A′ = ε ijk (λa1i )a2 j a3k = λ (ε ijk a1i a2 j a3k ) = λ A RVVLD LO GHWHUPLQDQWH A′ ULVXOWD SDUL D TXHOOR GL A SUHPROWLSOLFDWR SHU λ $QDORJD GLPRVWUD]LRQH QHO FDVR LQ FXL YHQJDQR SUHPROWLSOLFDWH ULJKH GLYHUVH GDOOD SULPD

'HWHUPLQDQWH GL XQD PDWULFH FRQ ULJKH FRORQQH XJXDOL

)LVVDWD XQD PDWULFH A VH XQD ULJD FRORQQD q XJXDOH DG XQ·DOWUD ULJD FRORQQD LO GHWHUPLQDQWH q QXOOR RVVLD A q XQD PDWULFH VLQJRODUH 6XSSRQLDPR FKH • OH ULJKH XJXDOL VLDQR DGLDFHQWL OD SULPD ULJD H OD VHFRQGD RSSXUH OD VHFRQGD H OD WHU]D ULJD VH VFDPELDPR OH GXH FRORQQH VL RWWLHQH XQD PDWULFH XJXDOH DOOD PDWULFH GL SDUWHQ]D RUD VDSSLDPR GD TXDQWR ULSRUWDWR QHO SDUDJUDIR FKH OR VFDPELR GL ULJKH FRPSRUWD FKH LO GHWHUPLQDQWH GHOOD PDWULFH RWWHQXWD GDOOR VFDPELR GHYH HVVHUH XJXDOH H GL VHJQR RSSRVWR DO GHWHUPLQDQWH GHOOD PDWULFH GL SDUWHQ]D PD SRLFKp OH GXH PDWULFL VRQR XJXDOL VHJXH FKH WDOH GHWHUPLQDQWH GHYH HVVHUH QXOOR • OH ULJKH XJXDOL QRQ VLDQR DGLDFHQWL RVVLD VLDQR XJXDOL OD SULPD H OD WHU]D VH VFDPELDPR OD SULPD ULJD FRQ OD WHU]D RWWHQLDPR XQD PDWULFH XJXDOH D TXHOOD GL SDUWHQ]D LO FXL GHWHUPLQDQWH GHYH HVVHUH XJXDOH PD GL VHJQR RSSRVWR D TXHOOR GHOOD PDWULFH GL SDUWHQ]D SHUWDQWR WDOH GHWHUPLQDQWH q QXOOR $QDORJDPHQWH QHO FDVR GHOOH FRORQQH

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a12 § a11 ¨ A = ¨ λa11 λa12 ¨a a32 © 31

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a12 a12 a32

a13 · ¸ a13 ¸ = λ A′ a33 ¸¹

(· FKLDUR TXLQGL FKH VXOOD PDWULFH A′ VLD KDQQR GXH ULJKH XJXDOL SHUWDQWR LO VXR GHWHUPLQDQWH q QXOOR 5LVXOWD TXLQGL QXOOR DQFKH LO GHWHUPLQDQWH GL A DQDORJDPHQWH QHO FDVR GL FRORQQH SURSRU]LRQDOL $QFKH LQ TXHVWR FDVR YDOJRQR FRQVLGHUD]LRQL JHRPHWULFKH DQDORJKH D TXHOOH VYLOXSSDWH QHO SDUDJUDIR $ WDO SURSRVLWR VL VXSSRQJD FKH OD PDWULFH A VLD OD PDWULFH GHL FRHIILFLHQWL GL XQ VLVWHPD OLQHDUH

­a11 x1 + a12 x 2 + a13 x 3 = b1 °° 1 2 3 ®a21 x + a22 x + a33 x = b2 ° 1 2 3 °¯a31 x + a32 x + a33 x = b3

OD FXL VROX]LRQH VH HVLVWH LQGLYLGXD LO SXQWR GL LQWHUVH]LRQH GHL WUH SLDQL UDSSUHVHQWDWL GDOOH WUH HTXD]LRQL GHO VLVWHPD 6H GXH ULJKH KDQQR L FRHIILFLHQWL SURSRU]LRQDOL D SULPR PHPEUR FRPH QHO VHJXHQWH HVHPSLR LQ FXL OD SULPD H OD WHU]D ULJD SUHVHQWDQR OD SURSRU]LRQDOLWj D SULPR PHPEUR

­a11 x1 + a12 x 2 + a13 x 3 = b1 °° 1 2 3 ®λa11 x + λa12 x + λa13 x = b2 ° 1 2 3 °¯a31 x + a32 x + a33 x = b3

VHJXH FKH A = 0 TXLQGL LO VLVWHPD QRQ DPPHWWH VROX]LRQL VL ULFRUGL FKH QHOOD VROX]LRQH GL XQ VLVWHPD OLQHDUH LO GHWHUPLQDQWH GHOOD PDWULFH GHL FRHIILFLHQWL VL WURYD D GHQRPLQDWRUH H TXLQGL VH q QXOOR LO VLVWHPD QRQ q ULVROYLELOH H FLz q FRHUHQWH FRQ LO IDWWR FKH OD SULPD H OD VHFRQGD HTXD]LRQH UDSSUHVHQWDQR GXH SLDQL SDUDOOHOL FKH QRQ SRVVRQR DYHUH SXQWL GL LQWHUVH]LRQH DO ILQLWR 9DOH DQFKH OD SURSULHWj LQYHUVD D TXHOOD GLPRVWUDWD QHO SUHVHQWH SDUDJUDIR RVVLD VH LO GHWHUPLQDQWH GL XQD PDWULFH A q QXOOR QHFHVVDULDPHQWH OD PDWULFH KD • ULJKH R FRORQQH FRQ L WHUPLQL WXWWL QXOOL LQ TXHVWR FDVR OD YHULILFD VL SXz HIIHWWXDUH VHPSOLFHPHQWH VYLOXSSDQGR LO GHWHUPLQDQWH VHFRQGR OD ULJD R OD FRORQQD QXOOD • DOPHQR GXH ULJKH FRQ L WHUPLQL SURSRU]LRQDOL 3HU GLPRVWUDUH O·DIIHUPD]LRQH GHO VHFRQGR SXQWR EDVWD DVVRFLDUH DOOD PDWULFH XQ VLVWHPD OLQHDUH AX = B FKH QHFHVVDULDPHQWH QRQ DPPHWWH VROX]LRQL SRLFKp OH GXH HTXD]LRQL FRQ L WHUPLQL D SULPR PHPEUR SURSRU]LRQDOL UDSSUHVHQWDQR SLDQL SDUDOOHOL H TXLQGL LO GHWHUPLQDQWH GL A GHYH HVVHUH QXOOR

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6LD A XQD PDWULFH H VL VXSSRQJD FKH XQD VXD ULJD VLD HVSUHVVD FRPH VRPPD GL GXH DGGHQGL RVVLD FKH O· i − esimo HOHPHQWR GHOOD p _ esima ULJD VLD GDWR GD

a pi = b pi + c pi (i = 1,2,3)

3DJ

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6L LQGLFKL FRQ Ab H Ac ULVSHWWLYDPHQWH OH PDWULFL FKH VL RWWHQJRQR GD

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]

A VRVWLWXHQGR OD p _ esima

ULJD FRQ OD ULJD (b pi ) H FRQ OD ULJD (c pi ) i = 1,2,3 YDOXWDQGR LO GHWHUPLQDQWH VL RWWLHQH

(

)

ε pqr A = ε ijk a pi aqj ark = ε ijk (b pi + c pi )aqj ark = = ε ijk b pi aqj ark + ε ijk c pi aqj ark = ε pqr Ab + ε pqr Ac

3RVVLDPR GXQTXH RWWHQHUH

ε pqr A = ε pqr Ab + ε pqr Ac ε pqr A = ε pqr ( Ab + Ac ) A = Ab + Ac

2VVLD LO GHWHUPLQDQWH GL A q SDUL DOOD VRPPD GHL GHWHUPLQDQWL GHOOD PDWULFH FRQ OH ULJKH FRVWLWXLWH GDL VLQJROL DGGHQGL 4XDQWR GHWWR SHU OH ULJKH YDOH DQFKH SHU OH FRORQQH H OD GLPRVWUD]LRQH SXz HVVHUH DSSOLFDWD LQ PRGR ULFRUVLYR DO FDVR GL ULJKH FRORQQH FRQ XQ QXPHUR TXDOVLDVL GL DGGHQGL H TXLQGL VXSSRQHQGR FKH O· i − esimo HOHPHQWR GHOOD p _ esima ULJD VLD GDWR GD n

a pi = ¦ b pi (i = 1,2,3) t =1

GHWWD

t

At OD PDWULFH FKH KD FRPH ULJD p _ esima OD ULJD IRUPDWD GDJOL HOHPHQWL (b pi ) SRVVLDPR t

VFULYHUH n

A = ¦ At t =1

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ik kj

k

­ °c1i = ¦ a1 p b pi p ° ° ®c2 j = ¦ a2 q bqj q ° °c = a b ° 3k ¦ 3r rk r ¯

§ ·§ ·§ · C = ε ijk c1i c 2 j c3k = ε ijk ¨ ¦ a1 p b pi ¸¨ ¦ a 2 q bqj ¸¨¨ ¦ a 3r brk ¸¸ ¨ p ¸¨ q ¸© r ¹ © ¹© ¹

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(

)(

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C = ε ijk ¦¦¦ a1 p b pi a2q bqj a3r brk = ε ijk ¦¦¦ a1 p a2 q a3r b pibqj brk p

q

r

p

3DJ

q

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C = ¦¦¦ a1 p a2 q a3r ε ijk b pi bqj brk = ¦¦¦ a1 p a2 q a3r ε pqr B p

q

r

p

q

r

ª º C = «¦¦¦ a1 p a2 q a3r ε pqr » B = A B «¬ p q r »¼

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A = A ⋅

A ⋅

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n volte

n

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LQ

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­a11 x1 + a12 x 2 + a13 x 3 = b1 °° 1 2 3 ®a21 x + a22 x + a33 x = b2 ° 1 2 3 ¯°a31 x + a32 x + a33 x = b3 ULSRUWDWR DOO·LQL]LR GHO SDUDJUDIR GRYH VL ULFRUGD FKH a LQGLFD LO GHWHUPLQDQWH GHOOD PDWULFH GHL FRHIILFLHQWL

A ­1 a22a33b1 − a23a32b1 − a12a33b2 + a12a23b3 + a13a32b2 − a13a22b3 °x = a11a22a33 − a11a23a32 − a12a21a33 + a12a23a31 + a13a21a32 − a13a22a31 ° ° 2 a11a33b2 − a11a23b3 − a21a33b1 + a23a31b1 + a13a21b3 − a13a31b2 ®x = a11a22a33 − a11a23a32 − a12a21a33 + a12a23a31 + a13a21a32 − a13a22a31 ° ° 3 a11a22b3 − a11a32b2 − a12a21b3 + a12a31b2 + a21a32b1 − a22a31b1 °x = a11a22a33 − a11a23a32 − a12a21a33 + a12a23a31 + a13a21a32 − a13a22a31 ¯

­ 1 1 ° x = a [(a22 a33 − a23a32 )b1 + (a13a32 − a12 a33 )b2 + (a12 a23 − a13 a22 )b3 ] ° ° 2 1 ® x = [(a23 a31 − a21a33 )b1 + (a11a33 − a13a31 )b2 + (a13a 21 − a11a23 )b3 ] a ° ° 3 1 ° x = a [(a21a32 − a22 a31 )b1 + (a12 a31 − a11a32 )b2 + (a11a 22 − a12 a21 )b3 ] ¯

§ a22 a33 − a23 a32 ¨ §x · ¨ a ¨ ¸ − a a a21a33 2 ¨ x ¸ = ¨ 23 31 a ¨¨ 3 ¸¸ ¨ ¨ x − a a a22 a31 © ¹ 21 32 ¨ a © 1

/D PDWULFH LQYHUVD q GXQTXH GDWD GD

a13 a32 − a12 a33 a a11a33 − a13 a31 a a12 a31 − a11a32 a

3DJ

a12 a23 − a13 a22 · ¸ a ¸§ b · a13 a21 − a11a23 ¸¨ b1 ¸ 2 ¸¨¨ ¸¸ a a11a22 − a12 a21 ¸© b3 ¹ ¸ a ¹

)RQGDPHQWL GL $OJHEUD &DSLWROR 'HWHUPLQDQWL

§ a22 a33 − a23a32 ¨ a ¨ a23a31 − a21a33 −1 ¨ A = ¨ a ¨ a21a32 − a22 a31 ¨ a ©

,QIDWWL X •

•

a13a32 − a12 a33 a a11a33 − a13 a31 a a12 a31 − a11a32 a

a12 a23 − a13a22 · ¸ a ¸ a13a21 − a11a23 ¸ ¸ a a11a22 − a12 a21 ¸ ¸ a ¹

= A −1 B AX = B H ULFRUGDQGR OD GHILQL]LRQH GL PDWULFH LQYHUVD VHJXH

AX = B AA −1 B = B AA −1 = I A −1 B = X A −1 B = A −1 AX = X A −1 A = I

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A−1 VL RWWLHQH GLYLGHQGR SHU LO GHWHUPLQDQWH A = a DOFXQL WHUPLQL

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(cij ) = §¨¨ aaps ©

rs

a pt · ¸ art ¸¹

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(c11 ) =

a22 a32

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a21 a13 (c32 ) = a31 a23

a11 a21

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LO ULVXOWDWR YLHQH PROWLSOLFDWR SHU (−1)

i+ j

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a11 a12 a13 a a a a a21 a22 a23 c12 = (−1)1+2 21 23 = 23 21 = a23a31 − a21a33 a31 a33 a33 a31 a31 a32 a33 •

a11 a12 a21 a22 a31

a32

a13 a a23 c31 = (−1)3 +1 12 a22 a33

a13 a23

= a12 a23 − a13a22

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a −1 ji =

cij

a a − a21a33 c12 −1 $G HVHPSLR a21 = 23 31 = a a

a

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AA−1 = I HVSOLFLWDQGR WDOH HVSUHVVLRQH VL KD

§ a11 1¨ ¨ a21 a¨ © a31

a13 ·§ c11 ¸¨ a23 ¸¨ c12 a33 ¸¹¨© c13

a12 a22 a32

1 AC T = I a

c21 c22 c23

c31 · § 1 0 0 · ¸ ¨ ¸ c32 ¸ = ¨ 0 1 0 ¸ c33 ¸¹ ¨© 0 0 1 ¸¹

HVHJXHQGR L SURGRWWL VL RWWLHQH FRQ

1 3 1 3 1 3 a1i c ji = δ1 j ¦ a2i c ji = δ 2 j ¦ a3i c ji = δ 3 j ¦ a i =1 a i=1 a i =1 FRQ •

j = (1,2,3)

1 3 ¦ aki c ji = δ kj a i=1

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a11 a12

a13

a a a a a21 a22 a23 = (−1)(1+1) a11 22 23 + (−1)(1+2) a12 21 23 + a32 a33 a31 a33 a31 a32 a33 + (−1)(1+3) a13

a21 a22 a31 a23

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c11 = (−1) (1+1)

a22 a32

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c13 = (−1) (1+3)

a21 a31

a22 a23

3

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( )−1 =

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δ ij = ®

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se

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2

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δ ij11ij22i3j3

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ε

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i i ....i

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i1i2 .....in

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