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
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.
last changed: January 19, 2005