On the second cohomology of semidirect products

TL;DR

This paper studies the second cohomology group of a semidirect product of groups, providing an exact sequence that relates it to the cohomology of its factors, and describes elements via extensions built from the factors.

Contribution

It introduces a natural five-term exact sequence connecting the second cohomology of a semidirect product with that of its subgroups, generalizing previous results.

Findings

Embedded the restriction map into a five-term exact sequence

Represented elements of H^2(G,M) via extensions from N and T

Connected cohomology of G with that of N and T through explicit constructions

Abstract

Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and a subgroup $T$, and let $M$ be a $G$-module with not necessarily trivial $G$-action. Then we embed the simultaneous restriction map $res=(res^G_N,res^G_T)^t : H^2(G,M) \to H^2(N,M)^T \times H^2(T,M)$ into a natural five term exact sequence consisting of one and two-dimensional cohomology groups of the factors $N$ and $T$. The elements of $H^2(G,M)$ are represented in terms of group extensions of $G$ by $M$ constructed from extensions of $N$ and $T$.