Extension of a group by a module

Question

What is the definition of an extension of a group by a module?

and what is the difference between extensions by groups and modules?

Answer 1

If $G$ is a group, then a $G$-module $V$ is an abelian group (also called $V$, their parents were not very creative) and a homomorphism $\phi:G \to \operatorname{Aut}(V)$. This is the same as being a $\mathbb{Z}[G]$-module, but perhaps the abelian group definition is more useful here.

A (downward) extension of $G$ by $V$ is a group $E$ with homomorphism $\iota:V\to E$ and $\pi:E \to G$ so that $\ker(\iota)=1$, $\operatorname{im}(\iota) = \ker(\pi)$, $\operatorname{im}(\pi)=G$, and $\pi^{-1}(g^{-1}) \iota(v) \pi^{-1}(g) = \iota( \phi(g)(v) )$ for all $g \in G$ and $v\in V$.

In simpler language, $V$ is a normal abelian subgroup of $E$ with quotient $G$, and the conjugation of $x \in E$ on $V$ is the same as the module action of $xV = g \in G$ on $V$.

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When $V$ is not an abelian group (but is still normal), then we still talk about extensions of $G$ by $V$ as being groups $E$ with $V$ a normal subgroup such that $E/V \cong G$, but the specific ways in which talk about the action of $G$ on $V$ change (in several contradictory ways, so if you don't need this level of generality, it is much better to ignore it until you do). The semi-direct product is an easy example where the action is still $\phi:G \to \operatorname{Aut}(V)$. However, if the extension is not a semidirect product, then the action is described by $\phi:G \to \operatorname{Aut}(V)/\operatorname{Inn}(V)$, but in this case some actions just don't work (they have an “obstruction” in $H^3(G,Z(V))$).