# Characteristic core

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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:

- (
*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. - (
*Automorph-intersection definition*): As the intersection of all automorphic subgroups to the given subgroup.

### Definition with symbols

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

- (
*Characteristic subgroup definition*): As the subgroup generated by all characteristic subgroups of such that ; in other words, as the unique largest characteristic subgroup of the whole group contained in the given group. - (
*Automorph-intersection definition*): As the intersection of all subgroups of of the form where is an automorphism of .