Group Extension - Knowlpedia

Let $N$ and $Q$ be groups. An extension of $Q$ by $N$ is a group $E$ together with a short exact sequence of groups

1 \longrightarrow N \xrightarrow{\;\iota\;} E \xrightarrow{\;\pi\;} Q \longrightarrow 1.

Exactness means $\iota$ is injective, $\pi$ is surjective, and $\iota(N)=\ker(\pi)$ .

In particular, $N$ identifies with a normal subgroup of $E$ , and $Q$ is (canonically) isomorphic to the quotient group $E/N$ . Extensions organize the ways a group can be built from a normal subgroup and a quotient.

Examples

For any $n\ge 1$ , there is an extension $1\to n\mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}\to 1$ .
The dihedral group fits into $1\to C_n\to D_{2n}\to C_2\to 1$ .
Direct products give extensions too: $1\to N\to N\times Q\to Q\to 1$ .