Central extension: Difference between revisions

Revision as of 01:46, 1 November 2011

A central extension is a group extension where the base normal subgroup is a central subgroup, i.e., it is contained in the center of the whole group.

Central extensions are significant because the central extensions for a given normal subgroup and quotient group are classified by the second cohomology group for trivial group action of the quotient group on the normal subgroup.

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 ==Definition==
-===Symbol-free definition===
+A '''central extension''' is a [[group extension]] where the base [[normal subgroup]] is a [[central subgroup]], i.e., it is contained in the [[center]] of the whole group.
-A '''central extension''' is an extension with quotient group being the given group, where the kernel is a [[central subgroup]].
+Central extensions are significant because the central extensions for a given [[normal subgroup]] and [[quotient group]] are classified by the [[second cohomology group for trivial group action]] of the quotient group on the normal subgroup.
-===Definition with symbols===
-A central extension of a group <math>G</math> is defined as a group <math>E</math> along with a surjective homomorphism <math>\rho:E \to G</math> such that the kernel of <math>\rho</math> is contained inside the [[center]] of <math>E</math>.
 ==Properties==