Editing Core (group theory) (section)

==The normal core==
===Definition===
For a group ''G'', the '''normal core''' or '''normal interior'''<ref>Robinson (1996) p.16</ref> of a subgroup ''H'' is the largest [[normal subgroup]] of ''G'' that is contained in ''H'' (or equivalently, the [[intersection (set theory)|intersection]] of the [[conjugate (group theory)|conjugates]] of ''H''). More generally, the core of ''H'' with respect to a [[subset]] ''S''&nbsp;&sube;&thinsp;''G'' is the intersection of the conjugates of ''H'' under ''S'', i.e.
:<math>\mathrm{Core}_S(H) := \bigcap_{s \in S}{s^{-1}Hs}.</math>

Under this more general definition, the normal core is the core with respect to ''S''&nbsp;=&thinsp;''G''. The normal core of any normal subgroup is the subgroup itself.

Dual to the concept of normal core is that of {{em|[[Normal closure (group theory)|normal closure]]}} which is the smallest normal subgroup of ''G'' containing ''H''.

===Significance===
Normal cores are important in the context of [[group action]]s on [[set (mathematics)|sets]], where the normal core of the [[isotropy subgroup]] of any point acts as the identity on its entire [[orbit (group theory)|orbit]]. Thus, in case the action is [[Transitive group action|transitive]], the normal core of any isotropy subgroup is precisely the [[kernel (algebra)|kernel]] of the action.

{{anchor|Core-free}}
A '''core-free subgroup''' is a subgroup whose normal core is the [[trivial subgroup]]. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive, [[Faithful group action|faithful]] group action.

The solution for the [[hidden subgroup problem]] in the [[Abelian group|abelian]] case generalizes to finding the normal core in case of subgroups of arbitrary groups.