The group homology 
of a group 
with coefficients in a right 
-module 
is the 
th left derived functor
of the coinvariants functor. The coinvariants are the
quotient group 
, where the denominator is
the additive subgroup generated by the indicated differences.
In particular, 
.
It may be computed by taking the module tensor
product over the group ring ![Z[G]](/images/equations/GroupHomology/Inline8.svg)
of 
with a projective resolution
of 
as a left ![Z[G]](/images/equations/GroupHomology/Inline11.svg)
-module with trivial 
-action, and taking the homology of the resulting chain
complex.
Group Homology
See also
Derived Functor, Group, Homology Group, Module, Projective ResolutionExplore with Wolfram|Alpha
References
Brown, K. S. Cohomology of Groups. New York: Springer-Verlag, 1982.Cite this as:
Weisstein, Eric W. "Group Homology." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupHomology.html