Cohomology of finite groups without homological algebra
  We prove explicit and elementary formulas for the group homology and cohomology of a finite group with coefficients in any module. We describe in elementary terms the cohomology algebra $H^*(G,k)$ as a graded algebra for a finite group $G$ and a commutative ring $k.$ 
Sergei O. Ivanov and Nikolay N. Mostovsky
math.GR