core of a subgroup

Let $H$ be a subgroup of a group $G$ .

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

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

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

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

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

Let $\cal L$ be the set of left cosets of $H$ in $G$ . By considering the action of $G$ on $\cal L$ it can be shown that the quotient (http://planetmath.org/QuotientGroup) $G/{\operatorname{core}}_{G}(H)$ embeds in the symmetric group $\operatorname{Sym}({\cal L})$ . A consequence of this is that if $H$ is of finite index in $G$ , then ${\operatorname{core}}_{G}(H)$ is also of finite index in $G$ , and $[G:{\operatorname{core}}_{G}(H)]$ divides $[G:H]!$ (the factorial of $[G:H]$ ). In particular, if a simple group $S$ has a proper subgroup of finite index $n$ , then $S$ must be of finite order dividing $n!$ , 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).

Title	core of a subgroup
Canonical name	CoreOfASubgroup
Date of creation	2013-03-22 15:37:22
Last modified on	2013-03-22 15:37:22
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	10
Author	yark (2760)
Entry type	Definition
Classification	msc 20A05
Synonym	core
Synonym	normal core
Synonym	normal interior
Related topic	NormalClosure2
Defines	core-free
Defines	corefree
Defines	normal-by-finite
Defines	core-free subgroup
Defines	corefree subgroup
Defines	normal-by-finite subgroup