Normal core

Symbol-free definition
The normal core of a subgroup of a group is defined in the following equivalent ways:


 * (Normal subgroup definition) As the subgroup generated by all defining ingredient::normal subgroups of the whole group lying inside the subgroup; in other words, the unique largest normal subgroup lying inside the given subgroup.
 * (Conjugate-intersection definition) As the intersection of all defining ingredient::conjugate subgroups of the given subgroup.
 * (Group action definition) As the kernel of the action of the group on the coset space of the subgroup by left multiplication (see group acts on left coset space of subgroup by left multiplication).

Definition with symbols
The normal core of a subgroup $$H$$ in a group $$G$$, denoted as $$H_G$$, is defined in the following equivalent ways:


 * (Normal subgroup definition) As the subgroup generated by all normal subgroups $$N$$ of $$G$$ that lie inside $$H$$ i.e.:

$$H_G := \langle N : N \triangleleft G, N \le H \rangle$$


 * (Conjugate-intersection definition) As the intersection of all subgroups of the form $$gHg^{-1}$$ where $$g$$ varies over $$G$$, i.e.:

$$H_G := \bigcap_{g \in G} gHg^{-1}$$


 * (Group action definition) As the kernel of the action of $$G$$ on the coset space $$G/H$$ (see group acts on left coset space of subgroup by left multiplication):

$$H_G = \operatorname{ker} (G \to Sym(G/H))$$

Other core operators in group theory

 * Characteristic core
 * Fully invariant core

Other operators involving approximating by normal subgroups

 * Normal closure is the smallest normal subgroup of the whole group containing a given subgroup.
 * Normalizer is the largest subgroup containing a given subgroup in which that subgroup is normal.

Analogous operators in other structures

 * Ideal core of a Lie subring: This is the analogous notion in Lie rings.
 * Normal closure of a field extension: This is the precise corresponding notion under the correspondence given by the fundamental theorem of Galois theory. Note that normal core becomes normal closure because of the direction-reversing nature of the correspondence.
 * Regular closure of a covering map: This is the precise corresponding notion under the correspondence between subgroups of the fundamental group and connected covers. Core becomes closure because of the direction-reversing nature of the correspondence.

Image
The normal core is an operator that takes subgroups to subgroups. It is an idempotent operator and the fixed points are precisely the normal subgroups.

In other words, the normal core of any subgroup is a normal subgroup, and the normal core of a normal subgroup is itself.

Inverse image of trivial subgroup
If the normal core of a subgroup is trivial, the subgroup is said to be a core-free subgroup.

Quotient by normal core
Note that because normality satisfies the intermediate subgroup condition, the normal core of a subgroup is normal within the subgroup. That is, if $$H$$ is a subgroup of $$G$$, $$H_G$$ is a normal subgroup of $$H$$. Thus, we can talk of the quotient $$H/H_G$$. The relation between properties of $$H$$ and properties of $$H/H_G$$ are studied in the theory of the quotient-by-core.

Computation
The normal core of a subgroup in a group can be found computationally by invoking the membership testing problem as well as the group intersection problem. The part involving invocation of the membership testing problem can be thought of as a variant of the normality testing problem.