Group Extension

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

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: $1\to N\to N\times Q\to Q\to 1$.