The group cohomology 
of a group 
with coefficients in a left 
-module 
is the 
th right derived functor
of the invariant-module functor 
. Here 
is an Abelian group with
an additive 
-action, and 
is its subgroup of group
fixed points.
In low dimensions, 
.
The cohomology group 
is the quotient group
of maps 
satisfying 
under pointwise addition,
modulo the subgroup of maps 
for a fixed 
. These are called derivations and principal derivations,
respectively (Weibel 1994, pp. 174-175). The cohomology
group 
classifies equivalence classes of group
extensions of 
by the additive Abelian group 
that induce the specified 
-action on 
. The equivalences induce the identity on both 
and 
(Weibel 1994, p. 183).
