6 1. Sylow Theory

1.5. Corollary. Let x G G, where G is a finite group, and let K be the

conjugacy class of G containing x. Then \K\ = \G :

CQ{X)\.

Proof. The class of x is the orbit of x under the conjugation action of G on

itself, and the stabilizer of x in this action is the centralizer

CQ(X).

Thus

|if | = \G :

C G ( # ) | ,

as required. •

1.6. Corollary. Let H C G be a subgroup, where G is finite. Then the total

number of distinct conjugates of H in G, counting H itself, is \G :

NG(H)\.

Proof. The conjugates of H form an orbit under the conjugation action of

G on the set of subsets of G. The normalizer NQ(H) is the stabilizer of H

in this action, and thus the orbit size is \G :

~NG(H)\,

as wanted. •

Problems 1A

1A.1. Let H be a subgroup of prime index p in the finite group G, and

suppose that no prime smaller than p divides |G|. Prove that H G.

1A.2. Given subgroups H,K C G and an element ^ G G , the set HgK —

{hgk \ h E H, k G K} is called an (ii, K)-double coset. In the case where

H and K are finite, show that \HgK\ = \H\\K\/\K n H\.

Hint. Observe that HgK is a union of right cosets of H, and that these

cosets form an orbit under the action of K.

Note. If we take g = 1 in this problem, the result is the familiar formula

\HK\ = \H\\K\/\HnK\.

1 A.3. Suppose that G is finite and that H^ K C G are subgroups.

(a) Show that \H \ H C\ K\ \G \ K\, with equality if and only if

HK = G.

(b) If \G : ii| and \G : K\ are coprime, show that HK — G.

Note. Proofs of these useful facts appear in the appendix, but we suggest

that readers try to find their own arguments. Also, recall that the product

HK of subgroups H and K is not always a subgroup. In fact, HK is a

subgroup if and only if HK — KH. (This too is proved in the appendix.)

If HK = KH, we say that H and K are permutable.

1A.4. Suppose that G — HK, where H and K are subgroups. Show that

also G =

HxKy

for all elements x,y G G. Deduce that if G —

HHX

for a

subgroup H and an element x G G, then H — G.