Stem extension: Difference between revisions

Revision as of 01:44, 1 November 2011

A stem extension' is a group extension where the base normal subgroup is contained in both the center and the derived subgroup of the whole group.

In particular, it is a special kind of central extension.

Schur covering group is a stem extension of maximal size. In the context of perfect groups, these are called universal central extensions.

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 ==Definition==
-===Definition with symbols===
+A ''stem extension''' is a [[defining ingredient::group extension]] where the base [[normal subgroup]] is contained in both the [[defining ingredient::center]] and the [[defining ingredient::derived subgroup]] of the whole group.
-A ''stem extension''' of a [[group]] <math>G</math> is 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 in the [[intersection of subgroups|intersection]] of the [[defining ingredient::center]] and the [[defining ingredient::derived subgroup]] of <math>E</math>.
+In particular, it is a special kind of [[defining ingredient::central extension]].
-A stem extension is a particular kind of [[defining ingredient::central extension]].
+==Related notions==
+* [[Schur covering group]] is a stem extension of maximal size. In the context of [[perfect group]]s, these are called [[universal central extension]]s.