The classification of finite groups acting freely on a homology 3-sphere is still incomplete. However it is known that these groups have periodic cohomology and groups with periodic cohomology are classified by the Suzuki–Zassenhaus theorem. The paper considers, more generally, finite groups which may act on a homology 3-sphere possibly with fixed points and the natural generalization of the Suzuki–Zassenhaus theorem to this case is found. According to the Suzuki–Zassenhaus theorem a group with periodic cohomology is either solvable or has a unique composition factor which is not cyclic, isomorphic to PSL(2, q) for q an odd prime. The main point of the work is that this fact is still true in the more general situation.