1 Answers
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q {\displaystyle Q} and N {\displaystyle N} are two groups, then G {\displaystyle G} is an extension of Q {\displaystyle Q} by N {\displaystyle N} if there is a short exact sequence
If G {\displaystyle G} is an extension of Q {\displaystyle Q} by N {\displaystyle N} , then G {\displaystyle G} is a group, ι {\displaystyle \iota } is a normal subgroup of G {\displaystyle G} and the quotient group G / ι {\displaystyle G/\iota } is isomorphic to the group Q {\displaystyle Q}. Group extensions arise in the context of the extension problem, where the groups Q {\displaystyle Q} and N {\displaystyle N} are known and the properties of G {\displaystyle G} are to be determined. Note that the phrasing " G {\displaystyle G} is an extension of N {\displaystyle N} by Q {\displaystyle Q} " is also used by some.
Since any finite group G {\displaystyle G} possesses a maximal normal subgroup N {\displaystyle N} with simple factor group G / N {\displaystyle G/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 N {\displaystyle N} lies in the center of G {\displaystyle G}.