Example 3

Analogously $H^2(G,M)$ is isomorphic to the $\mathbf{Z}$-module $Z^2(G,M)/B^2(G,M)$, where $Z^2(G,M)$ is the $\mathbf{Z}$-module of mappings $f$ of $G\times G$ into $M$, satisfying:

whatever $g_1,g_2,g_3$ in $G$ may be, and $B^2(G,M)$ is the sub-$\mathbf{Z}$-module of $Z^2(G,M)$ consisting of the $f$ for which there exists a mapping $h$ of $G$ into $M$ such that:

whatever $g_1,g_2$ in $G$ may be.

Thus one recovers the definition of the group $H^2(G,M)$ given in VIII, App. II; in particular there exists a canonical isomorphism of $H^2(G,M)$ onto the group of extension classes of $G$ by $M$ (loc. cit.).