The permutation character |

The next remark helps considerably to shorten the calculations necessary
for applications of this lemma. It shows that we can replace the summation
over all *g ÎG* by a summation over a *transversal*
of the conjugacy classes,
as the number of fixed points turns out to be constant on each such
class:

Lemma:The mappingis a bijection, and henceX_{g'}-> X_{gg'g-1}:x -> gxis ac:G ->N:g -> | X_{g}|class function, i.e. it is constant on the conjugacy classes ofG. More formally, for anyg,g' ÎG, we have that| X._{g'}| = | X_{gg'g-1}|

Proof: That *x -> gx* establishes a bijection between *X _{g'}* and

g'x = x iff gg'g^{-1}(gx) = gx.

The mapping
* c*
is called the *character*

of the action of *G* on *X*, or of * _{G}X*, in short.

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

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

The permutation character |