Definition 1

Let $F$ and $G$ be two groups. An extension of $G$ by $F$ is a triple $\mathscr{E} = (E, i, p)$, where $E$ is a group, $i$ an injective homomorphism of $F$ into $E$ and $p$ a surjective homomorphism of $E$ onto $G$ such that $\operatorname{Im}(i) = \operatorname{Ker}(p)$. A homomorphism $s : G \to E$ (resp. $r : E \to F$) such that $p \circ s = \operatorname{Id}_G$ (resp. $r \circ i = \operatorname{Id}_F$) is called a section (resp. retraction) of the extension $\mathscr{E}$.

An extension $\mathscr{E} = (E, i, p)$ of $G$ by $F$ is often denoted by the diagram $\mathscr{E} : F \xrightarrow{i} E \xrightarrow{p} G$, in which $i$ and $p$ are sometimes omitted if no confusion can arise. It is sometimes said simply that the group $E$ is an extension of $G$ by $F$.

For a group $E$ to be an extension of $G$ by $F$, it is necessary and sufficient that it contain a normal subgroup $F'$ isomorphic to $F$ such that the quotient group $E/F'$ is isomorphic to $G$.

An extension $\mathscr{E} : F \xrightarrow{i} E \xrightarrow{p} G$ is called central if the image $i(F)$ is contained in the centre of $E$; this is only possible if $F$ is commutative.

Let $\mathscr{E} : F \xrightarrow{i} E \xrightarrow{p} G$ and $\mathscr{E}' : F \xrightarrow{i'} E' \xrightarrow{p'} G$ be two extensions of $G$ by $F$. A morphism of $\mathscr{E}$ into $\mathscr{E}'$ is a homomorphism $u : E \to E'$ such that $p' \circ u = p$ and $u \circ i = i'$, or, in other words, such that the following diagram is commutative: