PlanetMath: core of a subgroup

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

core of a subgroup

(Definition)

Let be a subgroup of a group .

The core (or normal interior, or normal core) of in is the intersection of all conjugates of in :

$\displaystyle {{\mathrm{core}}}_{G}(H) = \bigcap_{x\in G}x^{-1}Hx.$

It is not hard to show that ${{\mathrm{core}}}_{G}(H)$ is the largest normal subgroup of contained in , that is, ${{\mathrm{core}}}_{G}(H)\trianglelefteq G$ and if $N\trianglelefteq G$ and $N\subseteq H$ then $N\subseteq{{\mathrm{core}}}_{G}(H)$ . For this reason, some authors denote the core by rather than ${{\mathrm{core}}}_{G}(H)$ , by analogy with the notation for the normal closure.

If ${{\mathrm{core}}}_{G}(H)=\{1\}$ , then is said to be core-free.

If ${{\mathrm{core}}}_{G}(H)$ is of finite index in , then is said to be normal-by-finite.

Let $\cal L$ be the set of left cosets of in . By considering the action of on $\cal L$ it can be shown that the quotient $G/{{\mathrm{core}}}_{G}(H)$ embeds in the symmetric group ${\mathrm{Sym}}({\cal L})$ . A consequence of this is that if is of finite index in , then ${{\mathrm{core}}}_{G}(H)$ is also of finite index in , and $[G:{{\mathrm{core}}}_{G}(H)]$ divides (the factorial of ). In particular, if a simple group has a proper subgroup of finite index , then must be of finite order dividing , as the core of the subgroup is trivial. It also follows that a group is virtually abelian if and only if it is abelian-by-finite, because the core of an abelian subgroup of finite index is a normal abelian subgroup of finite index (and the same argument applies if `abelian' is replaced by any other property that is inherited by subgroups).

"core of a subgroup" is owned by yark.

(view preamble)