Fully invariant core

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 $H$ of a group $G$ is defined in the following equivalent ways:

1. It is the join of all fully invariant subgroups of $G$ contained in $H$.
2. It is the set of $x\in H$ such that $\sigma (x)\in H$ for all endomorphisms $\sigma$ of $G$.
3. It is the intersection of all the subgroups $\sigma ^{-1}(H)$ for all endomorphisms $\sigma$ of $G$.