Group cocycle

A group cocycle is an element of the kernel of the coboundary map in the standard cochain complex computing group cohomology.

For a $G$-module $M$, a one-cocycle is a map $f:G\to M$ satisfying $f(gh)=f(g)+g\cdot f(h)$.

For a $G$-module $M$, a two-cocycle is a map $c:G\times G\to M$ satisfying