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) simeq G/G_{X}.

In the case when *G _{X}= {1 }*, the action is
said to be

A very trivial example is the *natural action*

of *S _{X}* on

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

harald.fripertinger "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

Actions of groups |