The d-Smith sets of Cartesian products of the alternating groups and finite elementary abelian 2-groups

Let G be a finite group. In 1970s, T. Petrie defined the Smith equivalence of real G-modules. The Smith set of G is the subset of the real representation ring consisting of elements obtained as differences of Smith equivalent real G-modules. Various results of the topic have been obtained. The d-Smith set of G is the set of all elements [V ]−[W] in the Smith set of G such that the H-fixed point sets of V and W have the same dimension for all subgroups H of G. The results of the Smith sets of the alternating groups and the symmetric groups are obtained by E. Laitinen, K. Pawa lowski and R. Solomon. In this paper, we give the calculation results of the d-Smith sets of the alternating groups and the symmetric groups. In addition, we give the calculation results of the d-Smith sets of Cartesian products of the alternating groups and finite elementary abelian 2-groups.

Mathematics Subject Classification. Primary 55M35, Secondary 20C15.