Characteristic core

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a subgroup operator related to the subgroup property characteristic subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

Definition

QUICK PHRASES: largest characteristic subgroup inside, biggest characteristic subgroup inside, intersection of all automorphs

Symbol-free definition

The characteristic core of a subgroup is defined in the following equivalent ways:

1. (Characteristic subgroup definition): As the subgroup generated by all characteristic subgroups of the whole group lying inside the subgroup; in other words, as the unique largest characteristic subgroup of the whole group contained in the given group.
2. (Automorph-intersection definition): As the intersection of all automorphic subgroups to the given subgroup.

Definition with symbols

The characteristic core of a subgroup $H$ of a group $G$ is defined in the following equivalent ways:

1. (Characteristic subgroup definition): As the subgroup generated by all characteristic subgroups $K$ of $G$ such that $K\leq H$; in other words, as the unique largest characteristic subgroup of the whole group contained in the given group.
2. (Automorph-intersection definition): As the intersection of all subgroups of $G$ of the form $\sigma (H)$ where $\sigma$ is an automorphism of $G$.