Group extension
Group for which a given group is a normal subgroup / From Wikipedia, the free encyclopedia
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In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If and are two groups, then is an extension of by if there is a short exact sequence
If is an extension of by , then is a group, is a normal subgroup of and the quotient group is isomorphic to the group . Group extensions arise in the context of the extension problem, where the groups and are known and the properties of are to be determined. Note that the phrasing " is an extension of by " is also used by some.[1]
Since any finite group possesses a maximal normal subgroup with simple factor group , 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 lies in the center of .