Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If ${\displaystyle Q}$ and ${\displaystyle N}$ are two groups, then ${\displaystyle G}$ is an extension of ${\displaystyle Q}$ by ${\displaystyle N}$ if there is a short exact sequence

${\displaystyle 1\to N\;{\overset {\iota }{\to }}\;G\;{\overset {\pi }{\to }}\;Q\to 1.}$

If ${\displaystyle G}$ is an extension of ${\displaystyle Q}$ by ${\displaystyle N}$, then ${\displaystyle G}$ is a group, ${\displaystyle \iota (N)}$ is a normal subgroup of ${\displaystyle G}$ and the quotient group ${\displaystyle G/\iota (N)}$ is isomorphic to the group ${\displaystyle Q}$. Group extensions arise in the context of the extension problem, where the groups ${\displaystyle Q}$ and ${\displaystyle N}$ are known and the properties of ${\displaystyle G}$ are to be determined. Note that the phrasing "${\displaystyle G}$ is an extension of ${\displaystyle N}$ by ${\displaystyle Q}$" is also used by some.^[1]

Since any finite group ${\displaystyle G}$ possesses a maximal normal subgroup ${\displaystyle N}$ with simple factor group ${\displaystyle G/\iota (N)}$, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups.

An extension is called a central extension if the subgroup ${\displaystyle N}$ lies in the center of ${\displaystyle G}$.