# 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 $G$:

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

It is not hard to show that
${\mathrm{core}}_{G}(H)$ is the largest normal subgroup^{} of $G$ contained in $H$,
that is, ${\mathrm{core}}_{G}(H)\mathrm{\u22b4}G$ and
if $N\mathrm{\u22b4}G$ and $N\subseteq H$ then $N\subseteq {\mathrm{core}}_{G}(H)$.
For this reason, some authors denote the core by ${H}_{G}$
rather than ${\mathrm{core}}_{G}(H)$,
by analogy^{} with the notation ${H}^{G}$ for the normal closure^{}.

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

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

Let $\mathcal{L}$ be the set of left cosets^{} of $H$ in $G$.
By considering the action of $G$ on $\mathcal{L}$ it can be shown that
the quotient^{} (http://planetmath.org/QuotientGroup) $G/{\mathrm{core}}_{G}(H)$ embeds in the symmetric group^{} $\mathrm{Sym}(\mathcal{L})$.
A consequence of this is that if $H$ is of finite index in $G$,
then ${\mathrm{core}}_{G}(H)$ is also of finite index in $G$,
and $[G:{\mathrm{core}}_{G}(H)]$ 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 |