# Fully invariant core

From Groupprops

This article defines a subgroup operator related to the subgroup property fully invariant 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

### Definition with symbols

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

- It is the join of all fully invariant subgroups of contained in .
- It is the set of such that for all endomorphisms of .
- It is the intersection of all the subgroups for all endomorphisms of .