# Characterisation of a class of group homomorphisms related to a central extension

Let $S$ and $R$ be groups and say $\sigma: S \twoheadrightarrow R$ is a group homomorphism that is a central extension; that is, it is surjective (extension) and its kernel is contained in the centre of $S$ (central). Let $\mathbf{ab} :\mathbf{Gp} \rightarrow\mathbf{Ab}$ be the abelianisation functor (the left adjoint to the inclusion of abelian groups into the category of groups). I would like to know of an elementary characterisation of the set of group homomorphisms $\phi: H \rightarrow R$ such that the canonical map $\Theta: S \times_R H \rightarrow S \times_{\mathbf{ab}S} \mathbf{ab}(S \times_R H)$ is an isomorphism.

Notes for the question

1. I am not assuming that $\phi$ is a surjection. If it were then Lemma 5.2.7 of 'Galois Theories' (Borceux and Janelidze) answers the question. (The answer is that $\Theta$ is iso. iff the commutator subgroups of $S$ and $S \times_R H$ are isomorphic.)

2. I know that if $\Theta$ is iso. then both $\phi$ and $\pi_2 : S \times_R H \rightarrow H$ are central, but the converse is false. A counterexample is to take $\phi$ to be $e:1 \rightarrow R$ for some $R$ with non-trivial commutator subgroup where $e$ is the identity of $R$.

3. If $\sigma=\phi$ and $\sigma$ is not necessarily central, then $\Theta$ is an isomorphism iff $\sigma$ is central.

I don't have the book "Galois Theories" at hand but it seems to me that the result of the Lemma you mention should hold even if $\phi$ is not surjective.
Indeed, given a commutative diagram $$\require{AMScd}\begin{CD}0 @>>> K @>{k}>> X @>{f}>> Y @>>> 0 \\ & @V{u}VV @V{v}VV @VV{w}V \\ 0 @>>> K' @>>{k'}> X' @>>{f'}> Y' @>>> 0 \end{CD} \tag{1}\label{1}$$ where the rows are short exact sequences, then $u$ is an isomorphism if and only if the canonical arrow $X\to X'\times_{Y'} Y$ is an isomorphism. Indeed, we have a diagram $$\begin{CD}0 @>>> K @>{k}>> X @>{f}>> Y @>>> 0 \\ & @V{u}VV @V{(v,f)}VV @VV{1_Y}V \\ 0 @>>> K' @>{(k',0)}>> X'\times_{Y'}Y @>{\psi_2}>> Y @>>> 0 \\& @V{1_{K'}}VV @V{\psi_1}VV @VV{w}V \\ 0 @>>> K' @>>{k'}> X' @>>{f'}> Y' @>>> 0, \end{CD}$$ where the middle row is exact; hence we can apply the Short Five Lemma to the the two upper rows.
Then it suffices to apply this to the case where the short exact sequences in \eqref{1} are given by the abelianizations of $S\times_HR$ and $S$.
• Thanks Arnaud. I may be misinterpreting what you are saying, but if you apply with abelianizations of $S \times_R H$ and $S$ respectively (i.e. $X=S \times_R H$, $X'=S$ etc) then the conclusion of the Short Five Lemma (en.wikipedia.org/wiki/Short_five_lemma) is that $S \times_R H \cong S$, which is not the condition. Oct 12, 2017 at 8:42