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Concept
Group extension
Summary
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.
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 .
One extension, the direct product, is immediately obvious. If one requires and to be abelian groups, then the set of isomorphism classes of extensions of by a given (abelian) group is in fact a group, which is isomorphic to
cf. the Ext functor. Several other general classes of extensions are known but no theory exists that treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem.
To consider some examples, if , then is an extension of both and . More generally, if is a semidirect product of and , written as , then is an extension of by , so such products as the wreath product provide further examples of extensions.
The question of what groups are extensions of by is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups , where each is an extension of by some simple group. The classification of finite simple groups gives us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general.
Official source
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