24 1. Sylow Theory

1.24. Corollary. Let P be a p-group of order

pa.

Then for every integer b

with 0 b a, there is a subgroup L P such that \L\ =

pb.

Proof. The assertion is trivial if b = 0, and so we can assume that b 0

and we work by induction on b. By the inductive hypothesis, there exists a

subgroup N P such that \N\ =

pb~x

and we can apply Theorem 1.22 (with

M — P) to produce a subgroup L P with \L : N\ — p. Then \L\ =

pb

and

the proof is complete. •

Recall now that the Sylow E-theorem can be viewed as a partial converse

to Lagrange's theorem. It asserts that for certain divisors k of an integer n,

every group of order n has a subgroup of order k. (The divisors to which

we refer, of course, are prime powers k such that n/k is not divisible by the

relevant prime.)

We can now enlarge the set of divisors for which we know that the

converse of Lagrange's theorem holds.

1.25. Corollary. Let G be a finite group, and suppose that

pb

divides \G\,

where p is prime and b 0 is an integer. Then G has a subgroup of order

Pb.

Proof. Let P be a Sylow p-subgroup of G and write \P\ =

pa.

Since

pb

divides |G|, we see that b a, and the result follows by Corollary 1.23. •

In fact, in the situation of Corollary 1.25, the number of subgroups of G

having order

pb

is congruent to 1 modulo p. (By Problem 1C.8 and the note

following it, it suffices to prove this in the case where G is a p-group, and

while this is not especially difficult, we have decided not to present a proof

here.) We mention also that it does not seem to be known whether or not

there are any integers n other than powers of primes such that every group

of order divisible by n has a subgroup of order n.

Sylow theory is also related to the theory of nilpotent groups in another

way: a finite group is nilpotent if and only if all of its Sylow subgroups are

normal. In fact, we can say more.

1.26. Theorem. Let G be a finite group. Then the following are equivalent.

(1) G is nilpotent.

(2)

~NG(H)

H for every proper subgroup H G.

(3) Every maximal subgroup of G is normal.

(4) Every Sylow subgroup of G is normal.

(5) G is the (internal) direct product of its nontrivial Sylow subgroups.