Group actions on spheres with rank one prime power isotropy

TL;DR

This paper characterizes when rank two finite groups can act on spheres with prime power isotropy, linking the existence of such actions to the absence of certain subgroup involvements.

Contribution

It provides a necessary and sufficient condition for these group actions, extending to a construction method via G-invariant representations on Sylow subgroups.

Findings

A finite G-CW-complex homotopy equivalent to a sphere exists under specific group conditions.

The main theorem connects group structure with the existence of sphere actions with prime power isotropy.

A construction method for G-CW-complexes is developed using representations on Sylow subgroups.

Abstract

We show that a rank two finite group G admits a finite G-CW-complex X homotopy equivalent to a sphere, with rank one prime power isotropy, if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a given G-invariant family of representations defined on the Sylow subgroups of G.