The Smith set of the group $S_{5} \times C_{2} \times \cdots \times C_{2}$

Abstract

In 1960, P.A. Smith raised an isomorphism problem. Is it true that the tangential $G$-modules at two fixed points of an arbitrary smooth $G$-action on a sphere with exactly two fixed points are isomorphic to each other? Given a finite group, the Smith set of the group means the subset of real representation ring consisting of all differences of Smith equivalent representations. Many researchers have studied the Smith equivalence for various finite groups. But the Smith sets for non-perfect groups were rarely determined. In particular, the Smith set for a non-gap group has not been determined unless it is trivial. We determine the Smith set for the non-gap group $G = S_{5} \times C_{2} \times \dots \times C_{2}$.