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

If $N$ has a finite resolution by finitely generated projective modules, restriction of the semidirect-product group ring gives a direct sum of copies of $\mathbb{Z}N$, indexed by the integers. On its kernel cohomology the cyclic generator shifts those copies, possibly twisting each copy by an automorphism. Invariants are zero because elements have finite support; coinvariants identify with a single copy. The two-column Lyndon–Hochschild–Serre spectral sequence therefore gives the displayed isomorphism of abelian groups. A group with such a finite resolution and cohomological dimension $d$ has $H^d(N;\mathbb{Z}N)\ne 0$, so the cyclic extension has dimension $d+1$.

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

For a normal subgroup $N$ of $G$ and a $G$-module $M$, the low-degree exact sequence begins $0\to H^1(G/N,M^N)\to H^1(G,M)\to H^1(N,M{)}^{G/N}\to H^2(G/N,M^N)$. The first map inflates cocycles along $G\to G/N$; the second restricts them. If the restriction is trivial, subtract a coboundary to make the cocycle vanish on $N$ and descend to the quotient. Thus for a finite Galois extension, the kernel of restriction on Galois cohomology with finite coefficients is finite.

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.

For a group extension $1\to H\to G\to Q\to 1$ and a $G$-module $M$, the transgressionis the first spectral sequence differential crossing from the vertical edge to the horizontal edge of the Lyndon–Hochschild–Serre spectral sequence. By the five-term exact sequence in group cohomology, an invariant one-cocycle class on $H$ has zero transgression precisely when it is the restriction of a one-cocycle class on $G$. Thus transgression measures an extension obstruction. An arbitrary extension of a representative to a group cochain on $G$ need not have a coboundary that already descends to $Q$; the cochain construction requires appropriate corrections.

For $1\to C_2\to C_4\to C_2\to 1$ with trivial coefficients ${\mathbb{F}}_2$, every group homomorphism $C_4\to {\mathbb{F}}_2$ kills its order-two subgroup. Consequently the nonzero class in $H^1(C_2,{\mathbb{F}}_2)$ has nonzero transgression. It detects this nonsplit group extension, represented by the carry two-cocycle $c(a,b)=ab$ for $a,b\in \{0,1\}$.

For the dihedral group $D_{10}=C_5⋊C_2$,The Lyndon–Hochschild–Serre spectral sequence proves this from the inversion action of $C_2$ on $H^2(C_5,\mathbb{Z})$.