Editing Group extension (section)

{{Short description|Group for which a given group is a normal subgroup}}
[[File:Group extension illustration.svg|thumb|400x400px|Extension of <math>Q</math> by <math>N</math>, resulting in the group <math>G</math>. They form a short exact sequence <math>1\to N\;\overset{\iota}{\to}\;G\;\overset{\pi}{\to}\;Q \to 1</math>. The [[Injective function|injective]] homomorphism <math>\iota</math> maps <math>N</math> to a normal subgroup of <math>G</math>. In turn, <math>\pi</math> maps <math>G</math> [[Surjective function|onto]] <math>Q</math>, sending each [[coset]] of <math>\iota(N)</math> to a different element of <math>Q</math>.]]
In [[mathematics]], a '''group extension''' is a general means of describing a [[group (mathematics)|group]] in terms of a particular [[normal subgroup]] and [[quotient group]].  If <math>Q</math> and <math>N</math> are two groups, then <math>G</math> is an '''extension''' of <math>Q</math> by <math>N</math> if there is a [[short exact sequence]]

:<math>1\to N\;\overset{\iota}{\to}\;G\;\overset{\pi}{\to}\;Q \to 1.</math>

If <math>G</math> is an extension of <math>Q</math> by <math>N</math>, then <math>G</math> is a group, <math>\iota(N)</math> is a [[normal subgroup]] of <math>G</math> and the [[quotient group]] <math>G/\iota(N)</math> is [[isomorphic]] to the group <math>Q</math>.  Group extensions arise in the context of the '''extension problem''', where the groups <math>Q</math> and <math>N</math> are known and the properties of <math>G</math> are to be determined. Note that the phrasing "<math>G</math> is an extension of <math>N</math> by <math>Q</math>" is also used by some.<ref>{{nlab|id=group+extension#Definition}} Remark 2.2.</ref>

Since any [[finite group]] <math>G</math> possesses a [[Maximal subgroup|maximal]] [[normal subgroup]] <math>N</math> with [[simple group|simple]] [[factor group]] <math>G/\iota(N)</math>, all finite groups may be constructed as a series of extensions with finite [[simple group]]s. This fact was a motivation for completing the [[classification of finite simple groups]].

An extension is called a '''central extension''' if the subgroup <math>N</math> lies in the [[center of a group|center]] of <math>G</math>.