ConstructionsTopWeightsMarks

Marks

We now consider another refinement (due to Burnside) of the Cauchy-Frobenius Lemma. It allows to enumerate orbits the elements of which have a given conjugacy class of subgroups as stabilizers. In particular it allows us to count orbits by their lengths. The problem is that for applications of this lemma we have to know certain matrices, the table of marks and its inverse, the calculation of which needs quite a good knowledge of the lattice L(G) of subgroups of G. Fortunately there are program systems at hand (like the Aachen subgroup lattice program and CAYLEY) that allow to treat a lot of nontrivial cases successfully.

Then we consider finite actions GX, where X is a poset and where G respects the order: x<x'Þgx<gx', i.e. G acts as a group of automorphisms on (X,£). This yields generalizations of several notions introduced in the preceding chapter and it gives further insight. In particular the Burnside ring will be introduced and we shall find an interesting explanation for the table of marks.


harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/
UNI-Graz Institut für Mathematik
UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

ConstructionsTopWeightsMarks