40 1. Sylow Theory

Proof. We have \G :

P\2

=

\G\2/\P\2

|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 =

24-32

for which 0

2

(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.

Problems IF

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

this.

Hint. Consider elements of the form

ra_1n_1mn,

where m G M and n G N.

Note. Recall that if x,y G G, then the element

x~1y~1xy

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.