## Actions of groups

Let *G* denote a multiplicative
group and *X* a nonempty set.
An *action*
of *G* on *X* is described by a mapping
*G ´X -> X :(g,x) -> gx, such that
g(g'x) = (gg')x, and 1x=x. *

We abbreviate this by saying that *G* *acts*
on *X* or simply by calling
*X* a *G*- *set* or by writing
_{G}X,

in short, since *G* acts from the left on *X*.
Before
we provide examples, we mention a second but equivalent formulation.
A homomorphism * d* from *G* into the *symmetric group*
group
symmetric
*S*_{X}:= { p | p:X -> X, bijectively }

on *X* is called a *permutation
representation of **G* on *X*.
It is easy to check
that the definition of action given above is equivalent to
* d:g -> bar (g), where bar (g) :x -> gx, is a permutation representation. *

The *kernel* of * d* will be
denoted by *G*_{X}, and so we have, if * bar (G):= d[G]*,
the isomorphism
*
bar (G) simeq G/G*_{X}.

In the case when *G*_{X}= {1 }, the action is
said to be *faithful*.
A very trivial example is the *natural action*
of *S*_{X} on *X* itself, where
the corresponding permutation representation
* d:p -> bar ( p)*
is the identity mapping. A number of less trivial examples will follow
in a moment.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001