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
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