On the table of marks of a direct product of finite groups

Abstract

The table of marks was first introduced by William Burnside in his book "Theory of groups of finite order" in 1955. The table of marks counts the number of fixed points one subgroup has in the action of the cosets of another. In doing this it also encodes a lot of useful information about the subgroup lattice of a group G, including the index of each of G's subgroups in both G and their normalizers, containments and what cyclic subgroups G has.

Despite their usefulness they are extremely expensive to compute (the GAP table of marks library extends only as far as the symmetric group on 13 letters). Thus one purpose of present research is find an efficient way to compute the table of marks of a direct product of finite group.

This is more difficult than one might expect. We consider a direct product of two finite groups GxH, using Goursat's lemma we hope to use knowledge of the table of marks of G and H to compute the table of marks of GxH. The methods developed in the present research also gives rise to a new base change matrix for the double Burnside algebra, QB(G, G), which it will be conjectured gives a cellular basis for the