Fully invariant core

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 defining ingredient::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 $$G$$.