Group extension

Description

A groups extension is a group which results from two other groups as quotient and normal subgroup.

Definition

Let $1\to N\stackrel{ı}{\to }G\stackrel{\pi }{\to }Q\to 1$ be a Short exact sequence. Then $N$ will be a Normal subgroup and $Q$ a Group quotient. $G$ is an extension of $Q$ by $N$.

Properties

Tip

Examples

Split extension

Let $1\to N\stackrel{ı}{\to }G\stackrel{\pi }{\to }Q\to 1$ be a Short exact sequence and $s:Q\to G$ be a Homomorphism splitting the short exact sequence. Then $G$ can be written as a Semi-direct product $N⋊Q$.