# Fully invariant core

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 $G$.