12 1. Sylow Theory

Proof. Let g G G. Then conjugation by g maps N onto itself, and it follows

that the restriction of this conjugation map to N is an automorphism of N.

(But note that it is not necessarily an inner automorphism of N.) Since K

is characteristic in TV, it is mapped onto itself by this automorphism of iV,

and thus

K9

— K, and it follows that K G. •

Problems IB

1B.1. Let S G Sylp(G), where G is a finite group.

(a) Let P C G be a p-subgroup. Show that PS is a subgroup if and

only if P C 5.

(b) If S G, show that Sylp(G) = {5}, and deduce that S is charac-

teristic in G.

Note. Of course, it would be "cheating" to do problems in this section using

theory that we have not yet developed. In particular, you should avoid using

the Sylow C-theorem, which asserts that every two Sylow p-subgroups of G

are conjugate in G.

IB.2. Show that Op(G) is the unique largest normal p-subgroup of G. (This

means that it is a normal p-subgroup of G that contains every other normal

p-subgroup of G.)

1B.3. Let S G Sylp(G), and write T V = NG(S), Show that N = TSSG(N).

IB.4. Let P C G be a p-subgroup such that \G : P\ is divisible by p. Using

Cauchy's theorem, but without appealing to Sylow's theorem, show that

there exists a subgroup Q of G containing P, and such that \Q : P\ — p.

Deduce that a maximal p-subgroup of G (which obviously must exist) must

be a Sylow p-subgroup of G.

Hint. Use Problem 1A.10 and consider the group

N G ( P ) / P .

Note. Once we know Cauchy's theorem, this problem yields an alternative

proof of the Sylow E-theorem. Of course, to avoid circularity, we appeal to

Problem 1A.8 for Cauchy's theorem, and not to Corollary 1.9.

IB.5. Let 7r be any set of prime numbers. We say that a finite group H is

a 7r-group if every prime divisor of \H\ lies in IT. Also, a 7r-subgroup H C G

is a Hall 7r-subgroup of G if no prime dividing the index \G : H\ lies in 7r.

(So if 7 T = {p}, a Hall 7r-subgroup is exactly a Sylow p-subgroup.)

Now let 6 : G — K be a surjective homomorphism of finite groups.

(a) If H is a Hall 7r-subgroup of G, prove that 0(H) is a Hall 7r-subgroup

ofK.