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.