Actions of groups

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

_GX,

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.

• Orbits
• Stabilizers
• Fixed points
• Examples
• Cosets
• The Cauchy-Frobenius Lemma
• The permutation character
• The Cauchy-Frobenius Lemma 2
• Similar Actions
• Exercises

Actions of groups