FREE ACTIONS OF FINITE GROUPS ON Sn x Sn

Let p be an odd prime. We construct a non-abelian extension Gamma of S-1 by Z/p x Z/p, and prove that any finite subgroup of Gamma acts freely and smoothly on S2p-1 x S2p-1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S2p-1 x S2p-1. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.