Group extension

Let $F$ and $G$ be groups. An extension of $G$ by $F$ is a solution to the problem of finding a group $E$ that contains a normal subgroup $F'$ isomorphic to $F$ such that the quotient group $E/F'$ is isomorphic to $G$ .

More specifically, an extension $\mathcal{E}$ of $F$ by $G$ is a triple $(E,i,p)$ where $E$ is a group, $i$ is an injective group homomorphism of $F$ into $E$ , and $p$ is a surjective homomorphism of $E$ onto $G$ such that the kernel of $p$ is the image of $i$ . Often, the extension $\mathcal{E}$ is written as the diagram $\mathcal{E} : F \stackrel{i}{\to} E \stackrel{p}{\to} G$ .

An extension is central if $i(F)$ lies in the center of $E$ ; this is only possible if $F$ is commutative. It is called trivial if $E = F \times G$ (the direct product of $F$ and $G$ ), $i$ is the canonical mapping of $F$ into $F\times G$ , and $p$ is the projection homomorphism onto $G$ .

Let $\mathcal{E} : F \stackrel{i}{\to} E \stackrel{p}{\to} G$ and $\mathcal{E}' : F \stackrel{i'}{\to} E' \stackrel{p'}{\to} G$ be two extensions of $G$ by $F$ . A morphism of extensions from $\mathcal{E}$ to $\mathcal{E}'$ is a homomorphism $f: E \to E'$ such that $f \circ i = i'$ and $p' \circ f = p$ .

A retraction of an extension is a homomorphism $r: E\to F$ such that $r\circ i$ is the identity function on $F$ . Similarly, a section of an extension is a homomorphism $s : G \to E$ such that $p \circ s = \text{Id}_G$ .

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Group extension