4 1. Sylow Theory

1.1. Theorem. Let H C G be a subgroup, and let ft be the set of right cosets

of H in G. Then

G/COY^Q{H)

is isomorphic to a subgroup ofSym(ft). In

particular, if the index \G : H\ — n, then G/coveciH) is isomorphic to a

subgroup of Sn, the symmetric group on n symbols.

Proof. The action of G on the set ft by right multiplication defines a

homomorphism 9 (the permutation representation) from G into Sym(O).

Since ker(#) = corec(i^), it follows by the homomorphism theorem that

G/covec{H) = 0(G), which is a subgroup of Sym(G). The last statement

follows since if \G : H| = n, then (by definition of the index) |fi| = n, and

thus Sym(fi) ^ Sn. •

1.2. Corollary. Let G be a group, and suppose that H C G is a subgroup

with \G : H\ — n. Then H contains a normal subgroup N of G such that

\G : N\ divides n\.

Proof. Take T V = corec(-ff). Then G/N is isomorphic to a subgroup of

the symmetric group Sn, and so by Lagrange's theorem, \G/N\ divides

\Sn\

= n\. •

1.3. Corollary. Let G be simple and contain a subgroup of index n 1.

Then \G\ divides n\.

Proof. The normal subgroup N of the previous corollary is contained in iJ,

and hence it is proper in G because n 1. Since G is simple, T V = 1, and

thus |G| = |G/JV| divides n\. •

In order to pursue our main goal, which is counting, we need to discuss

the "orbits" of an action. Suppose that G acts on O, and let a G ft. The

set Oa — {a*g \ g G G} is called the orbit of a under the given action. It is

routine to check that if (3 G 0

a

, then Op = Oa, and it follows that distinct

orbits are actually disjoint. Also, since every point is in at least one orbit,

it follows that the orbits of the action of G on ft partition ft. In particular,

if ft is finite, we see that \ft\ — $^ |C?|, where in this sum, O runs over the

full set of G-orbits on ft.

We mention some examples of orbits and orbit decompositions. First, if

H C G is a subgroup, we can let H act on G by right multiplication. It is

easy to see that the orbits of this action are exactly the left cosets of H in

G. (We leave to the reader the problem of realizing the right cosets of H

in G as the orbits of an appropriate action of H. But be careful: the rule

x*h = hx does not define an action.)

Perhaps it is more interesting to consider the conjugation action of G

on itself, where the orbits are exactly the conjugacy classes of G. The fact