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.
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