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