请输入您要查询的字词:

 

单词 CoreOfASubgroup
释义

core of a subgroup


Let H be a subgroupMathworldPlanetmathPlanetmath of a group G.

The core (or normal interior, or normal core) of H in Gis the intersectionMathworldPlanetmath of all conjugates of H in G:

coreG(H)=xGx-1Hx.

It is not hard to show thatcoreG(H) is the largest normal subgroupMathworldPlanetmath of G contained in H,that is, coreG(H)G andif NG and NH then NcoreG(H).For this reason, some authors denote the core by HGrather than coreG(H),by analogyMathworldPlanetmath with the notation HG for the normal closurePlanetmathPlanetmath.

If coreG(H)={1},then H is said to be core-free.

If coreG(H) is of finite index in H,then H is said to be normal-by-finite.

Let be the set of left cosetsMathworldPlanetmath of H in G.By considering the action of G on it can be shown thatthe quotientPlanetmathPlanetmath (http://planetmath.org/QuotientGroup) G/coreG(H) embeds in the symmetric groupMathworldPlanetmathPlanetmath Sym().A consequence of this is that if H is of finite index in G,then coreG(H) is also of finite index in G,and [G:coreG(H)] divides [G:H]! (the factorialMathworldPlanetmath of [G:H]).In particular, if a simple groupMathworldPlanetmathPlanetmath S has a proper subgroupMathworldPlanetmath of finite index n,then S must be of finite order dividing n!,as the core of the subgroup is trivial.It also follows thata group is virtually abelian if and only if it is abelian-by-finite,because the core of an abelianMathworldPlanetmath subgroup of finite indexis a normal abelian subgroup of finite index(and the same argument applies if ‘abelian’ is replaced byany other property that is inherited by subgroups).

Titlecore of a subgroup
Canonical nameCoreOfASubgroup
Date of creation2013-03-22 15:37:22
Last modified on2013-03-22 15:37:22
Owneryark (2760)
Last modified byyark (2760)
Numerical id10
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20A05
Synonymcore
Synonymnormal core
Synonymnormal interior
Related topicNormalClosure2
Definescore-free
Definescorefree
Definesnormal-by-finite
Definescore-free subgroup
Definescorefree subgroup
Definesnormal-by-finite subgroup
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 8:36:11