Most Rank Two Finite Groups Act Freely on a Homotopy Product of Two Spheres

A classic result of Swan states that a finite group acts freely on a homotopy sphere if and only if every abelian subgroup of is cyclic. Following this result, Benson and Carlson conjectured that a finite group acts freely on the homotopy product of spheres if the rank of is less than . Recently Adem and Smith have shown that every rank two finite -group acts on a homotopy product of two spheres. In this paper we will make further progress on showing that rank two groups act freely on a homotopy product of two spheres. Letting be the semidirect product where the action is given by the obvious inclusion , we will show that a rank two finite group acts freely on a homotopy product of two spheres if it does not contain a subgroup isomorphic to for some prime