Lyndon–Hochschild–Serre spectral sequence

For a normal subgroup $H◃G$, quotient $Q=G/H$, and a $G$-module $M$, the Lyndon–Hochschild–Serre spectral sequence has

The low-degree edge maps of the Lyndon–Hochschild–Serre spectral sequence give

Inflation composes a cocycle on the quotient $Q=G/H$ with the quotient map $G\to Q$ and regards its values in the invariant submodule $M^H$ as values in $M$.

Restriction sends a cocycle on $G$ to its restriction to a subgroup $H$. If $H$ is normal, its image in $H^1(H,M)$ is fixed by the induced $G/H$-action.

The transgression $d_2:H^1(H,M{)}^Q\to H^2(Q,M^H)$ is the first differential crossing from the vertical to the horizontal edge of the Lyndon–Hochschild–Serre spectral sequence. If a representative on $H$ is extended to a cochain on $G$, its coboundary descends to the representing two-cocycle on $Q$.