Endomorphisms in short exact sequences

Abstract

We sudy the behaviour of endomorphisms and automorphisms of groups involved in abelian group extensions. The main result can be stated as follows: Let $0\to N\to G\to Q \to 1$ be an abelian group extension. Then one has the following exact sequence of groups: $$0\to End^{N,Q}(G)\to End^Q_N(G)\to End_Q(N)\to H^2(Q,N)\to H^2(G,N)$$ where $End^{N,Q}(G)$ denotes the set of all endomorphisms of $G$ which centralise $N$ and induce identity on $Q$, $End^Q_N(G)$ denotes the set of all endomorphisms of $G$ which normalise $N$ and induce identity on $Q$ and $End_Q(N)$ denotes the set of endomorphisms of $N$ which are compatible with the action of $Q$ on $N$. This exact sequence is obtained using the five-term exact sequence in group cohomology. An interesting fact we discovered is that the first three terms involved have ring structure and the maps between them are ring homomorphisms. The ring structure on $End_Q(N)$ is well-known, however the ring structure of the second term is a little more exotic. Restricted on quasi-regular elements, this gives the exact sequence proved recently in \cite{passi} by Passi, Singh and Yadav.