It is proved that the homology and cohomology theories of groups and associative algebras are non-abelian derived functors of the cokernel and kernel groups of higher dimensions of their defining standard chain and cochain complexes respectively. The same results are also obtained for the relative (co)homology of groups, the mod q cohomology of groups and the cohomology of groups with operators. This allowed us to give an alternative approach to higher Hopf formulas for integral homology of groups. An axiomatic characterization of the relative cohomology of groups is given and higher relative $(n+1)$-th cohomology of groups is described in terms of n-fold extensions.
"More about the (co)homology of groups and associative algebras." Homology Homotopy Appl. 7 (1) 87 - 108, 2005.