Mod qcohomology and Tate–Vogel cohomology of groups

The notions of mod qcohomology and Tate–Farrell–Vogel cohomology of groups are introduced, where q is a positive integer. The first and the second mod qcohomology groups are described in terms of torsors and extensions respectively. The mod qcohomology of groups is expressed as cotriplecohomology. The reduction of mod q Tate–Farrell–Vogel cohomology theory to the case q=pm with p a prime is shown. For finite groups with periodic cohomology the periodicity of mod q Tate cohomology is proved.