1A 5

that for a finite group, the order \G\ is the sum of the sizes of the classes is

sometimes called the class equation of G.

How big is an orbit? The key result here is the following.

1.4. Theorem (The Fundamental Counting Principle). Let G act on Q,

and suppose that O is one of the orbits. Let a G O, and write H = Ga, the

stabilizer of a. Let A = {Hx \ x G G} be the set of right cosets of H in G.

Then there is a bisection 0 : A — O such that 0{Hg) = a*g. In particular,

\0\ = \G:Ga\.

Proof. We observe first that if Hx = Hy, then

OL*X

— a»y. To see why this

is so, observe that we can write y — hx for some element h G H. Then

a*y = a*(hx) = (a*h)*x = a*x,

where the last equality holds because h G H = Ga, and so h stabilizes a.

Given a coset Hx G A, the point a*x lies in (9, and we know that it is

determined by the coset Hx, and not just by the particular element x. It is

therefore permissible to define the function 0 : A — O by 0(Hx) = a*x, and

it remains to show that 0 is both injective and surjective.

The surjectivity is easy, and we do that first. If j3 G O, then by the

definition of an orbit, we have f3 — a*x for some element x G G. Then

Hx G A satisfies 0(Hx) = a-x = /?, as required.

To prove that 0 is injective, suppose that 0(Hx) — 0{Hy). We have

a'X = a-y, and hence

a = a«l =

(a*x)*x~l

—

(a*y)-x~1

— a*{yx~ ) .

Then

yx'1

fixes a, and so it lies in Ga = H. It follows that y G Hx, and

thus Hy = Hx. This proves that 0 is injective, as required. •

It is easy to check that the bijection 0 of the previous theorem actually

defines a "permutation isomorphism" between the action of G on A and the

action of G on the orbit O. Formally, this means that 0(X*g) — 0(X)*g

for all "points" X in A and group elements g G G. More informally, this

says that the actions of G on A and on O are "essentially the same". Since

every action can be thought of as composed of the actions on the individual

orbits, and each of these actions is permutation isomorphic to the right-

multiplication action of G on the right cosets of some subgroup, we see that

these actions on cosets are truly fundamental: every group action can be

viewed as being composed of actions on right cosets of various subgroups.

We close this section with two familiar and useful applications of the

fundamental counting principle.