core of a subgroup

Let H be a subgroupMathworldPlanetmathPlanetmath of a group G.

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


It is not hard to show that coreG(H) is the largest normal subgroupMathworldPlanetmath of G contained in H, that is, coreG(H)G and if NG and NH then NcoreG(H). For this reason, some authors denote the core by HG rather 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 that the quotientPlanetmathPlanetmath ( 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 that a group is virtually abelian if and only if it is abelian-by-finite, because the core of an abelianMathworldPlanetmath 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