Note that ( s, r 2 ) and ( r4 ) are precisely the j oin and intersection, respectively, of these two subgroups in D t 6 · Finally, given the lattice of subgroups of a group, it is relatively easy to compute normalizers and centralizers. For example, in Ds we can see that C Ds (s) = ( s, r 2 ) be cause we first calculate that r 2 E C Ds (s) (see Section 2). This proves ( s, r 2 ) _:::: C Ds (s) (note that an element always belongs to its own centralizer) . The only subgroups which contain ( s, r 2 ) are that subgroup itself and al l o f D8 . We cannot have CDs (s) = D8 because r does not commute with s (i e . , r ¢ CDs (s) ) . This leaves only the claimed possibility for CDs (s). .
EX E RC I S E S 1. Let H and K be subgroups of G. Exhibit all possible sublattices which show only G, 1 , H, K and their joins and intersections. What distinguishes the different drawings? 2. In each of (a) to (d) list all subgroups of D t 6 that satisfy the given condition. 4 (a) Subgroups that are contained in ( sr 2 , r ) 4 (b) Subgroups that are contained in ( sr 7, r ) (c) Subgroups that contain ( r4 ) (d) Subgroups that contain ( s ) .
3. Show that the subgroup ( s, r 2 ) of D8 is isomorphic to V4 .
4.
Use the given lattice to find all pairs of elements that generate Ds (there are 1 2 pairs).
5. Use the given lattice to find all elements such elements x ) .
x
E Dt 6 such that D t6
=
( x, s ) (there are 16
6. Use the given lattices to help find the centralizers of every element in the following groups: (d) D t 6 · (b) Qs (c) S3 (a) Ds 7. Find the center of D t6· 8. In each of the following groups find the normalizer of each subgroup: (a)
S3
(b) Qs .
9. Draw the lattices of subgroups of the following groups: (a) Z/ 1 6/Z (b) Z/24/Z (c) Zj48Z. [See Exercise 6 in Section 3.]
10. Classify groups of order 4 by proving that if I G I = Exercise 36, Section L . l .]
4 then
G � Z4 or G � V4 . [See
11. Consider the group of order 16 with the following presentation: Q D1 6 =
( a, T I a 8 = T 2
= 1,
aT
=
Ta 3 )
(called the quasidihedral or semidihedral group of order 1 6). This group has three sub groups of order 8: ( T, a 2 ) � D8 , ( a ) � Z8 and ( a 2 , a T ) � Q8 and every proper subgroup is contained in one of these three subgroups. Fill in the missing subgroups in the lattice of all subgroups of the quasidihedral group on the following page, exhibiting each subgroup with at most two generators. (This is another example of a nonplanar lattice.) The next three examples lead to two nonisomorphic groups that have the same lattice of sub groups. 12. The group A = z2 X Z4 = ( a , b I a 2 = b = 1 , ab = ba ) has order 8 and has three subgroups of order 4: ( a , b2 ) � V4 , ( b ) � Z4 and ( ab ) � Z4 and every proper
4
Sec. 2 . 5
The lattice of Su bgroups of a Group
71