2 1. Sylow

(1) a«l — a for all a G ft and

(2) (a*g)'h — a*(gh) for all a G fi and all group elements g}h G G.

Suppose that G acts on ft. It is easy to see that if g G G is arbitrary,

then the function ag : fi — O defined by

(aOa#

= a-g has an inverse: the

function jp-i. Therefore, ag is a permutation of the set ft, which means that

Gg is both injective and surjective, and thus ag lies in the symmetric group

Sym(fi) consisting of all permutations of ft. In fact, the map g t-^ ag is

easily seen to be a homomorphism from G into Sym(fJ). (A homomorphism

like this, which arises from an action of a group G on some set, is called a

permutation representation of G.) The kernel of this homomorphism is,

of course, a normal subgroup of G, which is referred to as the kernel of the

action. The kernel is exactly the set of elements g G G that act trivially on

ft, which means that ocg — a for all points a G ft.

Generally, we consider a theorem or a technique that has the power

to find a normal subgroup of G to be "good", and indeed permutation

representations can be good in this sense. (See the problems at the end of

this section.) But our goal in introducing group actions here is not to find

normal subgroups; it is to count things. Before we proceed in that direction,

however, it seems appropriate to mention a few examples.

Let G be arbitrary, and take ft = G. We can let G act on G by right

multiplication, so that x*g — xg for x,g E G. This is the regular action of

G, and it should be clear that it is faithful, which means that its kernel is

trivial. It follows that the corresponding permutation representation of G is

an isomorphism of G into Sym(G), and this proves Cayley's theorem: every

group is isomorphic to a group of permutations on some set.

We continue to take ft = G, but this time, we define x*g —

g~xxg.

(The

standard notation for

g~lxg

is

x9.)

It is trivial to check that

x1

= x and that

(x9)h

=

xgh

for all x,g,h G G, and thus we truly have an action, which is

called the conjugation action of G on itself. Note that

x9

— x if and only if

xg = gx, and thus the kernel of the conjugation action is the set of elements

g G G that commute with all elements x G G. The kernel, therefore, is the

center Z(G).

Again let G be arbitrary. In each of the previous examples, we took

O = G, but we also get interesting actions if instead we take ft to be the set

of all subsets of G. In the conjugation action of G on ft we let X-g =

X9

—

{x9

| x G X} and in the right-multiplication action we define X*g — Xg =

{xg | x G X}. Of course, in order to make these examples work, we do not

really need ft to be all subsets of G. For example, since a conjugate of a

subgroup is always a subgroup, the conjugation action is well defined if we

take ft to be the set of all subgroups of G. Also, both right multiplication