40 1. Sylow Theory
Proof. We have \G :
|G|, and thus Op(G) 1 by
Corollary 1.39. •
We mention that the conclusion of Brodkey's theorem definitely can fail
if the Sylow p-subgroups are nonabelian. A counterexample is a certain
group G of order 144 =
for which 0
(G) = 1. Taking p = 2, we
see that the conclusion of Corollary 1.40 fails for this group, and thus the
conclusion of Brodkey's theorem must also fail. Of course, the Sylow 2-
subgroups of G are necessarily nonabelian in this case.
This counterexample of order 144 can be constructed as follows. Let E
be elementary abelian of order 9. (In other words, E is the direct product
of two cyclic groups of order 3.) Then E can be viewed as a 2-dimensional
vector space over a field of order 3, and thus the full automorphism group
of E is isomorphic to GL(2,3), which has order 48. A Sylow 2-subgroup
T of this automorphism group, therefore, has order 16, and of course, since
T C Aut(E), we see that T acts on E and that this action is faithful.
Using the "semidirect product" construction, which we discuss in Chap-
ter 3, we can build a group G containing (isomorphic copies of) E and T,
where E\G — ET, and the conjugation action of T on E within G is identi-
cal to the original faithful action of T on E. Now 02(G) G and E\G, and
since 02(G) is a 2-group and E is a 3-group, we see that 02(G) D E = 1. It
follows that 02(G) centralizes E, and thus 02(G) is contained in the kernel
of the conjugation action of T on E. This action is faithful, however, and
we deduce that 02(G) = 1, and indeed we have a counterexample.
1F.1. In the last paragraph of this section, we used the fact that if M G
and N G and M fl T V = 1, then M and N centralize each other. Prove
Hint. Consider elements of the form
where m G M and n G N.
Note. Recall that if x,y G G, then the element
G G is the
commutator of x and y. It is customary to write [x,y] to denote this
element. Note that [x, y] — 1 if and only if xy — yx. We study commutators
in more detail in Chapter 4.
IF.2. Let G be a group, and fix a prime p. Show that if Op(G) = 1, then
there exist S,T G Sylp(G) such that Z(5) D Z(T) = 1.
1F.3. Let G = NP, where NG,P G Sylp(G) and Nf)P = 1, and assume
that the conjugation action of P on N is faithful. Show that P acts faithfully
on at least one orbit of this action.