Normal core

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In mathematics, the normal core of a subgroup of a group is the largest normal subgroup of the group contained inside that subgroup. Equivalently, it is the intersection of the conjugates of the given subgroup.

The normal core of any normal subgroup is the subgroup itself.

Normal cores become important in the context of group actions on sets, where the normal core of the isotropy subgroup on any point acts as the identity on its entire orbit. Thus, in case the action is transitive, the normal core of any isotropy subgroup is precisely the kernel of the action.

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.

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