1A

3

and conjugation preserve cardinality, and so each of these actions makes

sense if we take ft to be the collection of all subsets of G of some fixed size.

In fact, as we shall see, the trick in Wielandt's proof of the Sylow existence

theorem is to use the right multiplication action of G on its set of subsets

with a certain fixed cardinality.

We mention one other example, which is a special case of the right-

multiplication action on subsets that we discussed in the previous paragraph.

Let H C G be a subgroup, and let ft = {Hx \ x G G}, the set of right cosets

of H in G. If X is any right coset of H, it is easy to see that Xg is also a

right coset of H. (Indeed, if X — Hx, then Xg = H(xg).) Then G acts on

the set ft by right multiplication.

In general, if a group G acts on some set ft and a G ft, we write Ga —

{g G G | a*g — a}. It is easy to check that Ga is a subgroup of G; it

is called the stabilizer of the point a. For example, in the regular action

of G on itself, the stabilizer of every point (element of G) is the trivial

subgroup. In the conjugation action of G on G, the stabilizer of x G G

is the centralizer CG(X) and in the conjugation action of G on subsets, the

stabilizer of a subset X is the normalizer

'NQ(X).

A useful general fact about

point stabilizers is the following, which is easy to prove. In any action, if

a-g = /?, then the stabilizers Ga and Gp are conjugate in G, and in fact,

(Gay =

GP.

Now consider the action (by right multiplication) of G on the right cosets

of i7, where H C G is a subgroup. The stabilizer of the coset Hx is the

set of all group elements g such that Hxg = Hx. It is easy to see that g

satisfies this condition if and only if xg G Hx. (This is because two cosets

Hu and Hv are identical if and only if u G Hv.) It follows that g stabilizes

Hx if and only if g G

x~xHx.

Since

x~xHx

—

Hx,

we see that the stabilizer

of the point (coset) Hx is exactly the subgroup

Hx,

conjugate to H via

x. It follows that the kernel of the action of G on the right cosets of H in

G is exactly f]

Hx.

This subgroup is called the core of H in G, denoted

x£G

coreciH). The core of H is normal in G because it is the kernel of an action,

and, clearly, it is contained in H. In fact, if N G is any normal subgroup

that happens to be contained in H, then N =

Nx

C

Hx

for all x G G, and

thus N C coreciH). In other words, the core of H in G is the unique largest

normal subgroup of G contained in H. (It is "largest" in the strong sense

that it contains all others.)

We have digressed from our goal, which is to show how to use group

actions to count things. But having come this far, we may as well state the

results that our discussion has essentially proved. Note that the following

theorem and its corollaries can be used to prove the existence of normal

subgroups, and so they might be considered to be "good" results.