The cohomology rings of the special affine group of and of PSL(3,Fp)



The special affine group of, p an odd prime, denoted Qd(p), plays an important role in the search for free actions by finite groups on products of spheres. The mod-p and integral cohomology rings of Qd(p) are computed and, as an extension of these results, the mod-p and p-primary part of the integral cohomology of the simple group PSL(3, p) are computed. Various properties of these rings are discussed. free action on a manifold using surgery-theoretic techniques on Swan’s free G-complex X. That is, they showed that the Smith and Milnor conditions were necesssary and sufficient.
It can be shown (see [16] Theorem XII.11.6) that the cohomology of a group G is periodic if and only if every abelian subgroup of G is cyclic, a condition which can be rephrased in terms of the rank of the group:
Definition 0.0.2. The rank of a finite group G, denoted r(G), is the maximum value of the p-ranks,
rp(G) = max{n|(Zp)n ,→ G},
for all p dividing the order of G.
A group of the form (Zp)n is called an elementary abelian group of rank n.
Hence, a group G is periodic if and only if r(G)=1. Since a rank-1 group acts freely on a finite complex having the homotopy type of one sphere, this led to the following conjecture (appearing in [7]):
Conjecture 0.0.5. A finite group of rank n acts freely on a finite complex homotopy equivalent to Sm1 ×···×Smn.
It should be noted that free actions arising from representations (called “linear spheres”) can always be used to find such free actions for p-groups, but cannot be used in general (see [36]).
We will outline the present state of research on this problem and, after some necessary background is given, explain the contributions that the present work makes. In particular, we will restrict attention to the problem of finding free actions by rank-2 finite groups on finite complexes homotopy equivalent to a product of two spheres.