10 1. Sylow Theory
Since these polynomials are congruent, the coefficients of corresponding
terms are congruent modulo p, and the result follows by considering the
coefficient of
Xp
on each side.
Proof of the Sylow E-theorem (Wielandt). Write \G\ =
pamJ
where
a 0 and p does not divide ra. Let fi be the set of all subsets of G having
cardinality p
a
, and observe that G acts by right multiplication on fi. Because
of this action, Vt is partitioned into orbits, and consequently, \Q\ is the sum
of the orbit sizes. But
(
J)a7Tl\
) = m jk 0 mod p,
pa J
and so \Q\ is not divisible by p, and it follows that there is some orbit O
such that \0\ is not divisible by p.
Now let X G (9, and let H = Gx be the stabilizer of X in G. By
the fundamental counting principle, \0\ \G\j\H\, and since p does not
divide \0\ and
pa
divides |G|, we conclude that
pa
must divide \H\, and in
particular^ \H\.
Since H stabilizes X under right multiplication, we see that if x G X,
then xH C X, and thus \H\ = \xH\ \X\ = p
a
, where the final equality
holds since X G Q. We now have \H\ =
pa,
and since H is a subgroup, it is
a Sylow subgroup of G, as wanted.
In Problem 1A.8, we sketched a proof of Cauchy's theorem. We can now
give another proof, using the Sylow E-theorem.
1.9. Corollary (Cauchy). Let G be a finite group, and suppose that p is a
prime divisor of \G\. Then G has an element of order p.
Proof. Let S be a Sylow p-subgroup of G, and note that since | *S^| is the
maximum power of p that divides |G|, we have |5| 1. Choose a non-
identity element x of 5, and observe that the order o(x) divides |5| by
Lagrange's theorem, and thus 1 o(x) is a power of p. In particular, we
can write o(x) pm for some integer m 1, and we see that
o(xm)
= p, as
wanted.
We introduce the notation Sylp(G) to denote the set of all Sylow p-
subgroups of G. The assertion of the Sylow E-theorem, therefore, is that
the set Sylp(G) is nonempty for all finite groups G and all primes p. The
intersection p|Sylp(G) of all Sylow p-subgroups of a group G is denoted
Op(G), and as we shall see, this is a subgroup that plays an important role
in finite group theory.
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