Fully invariant closure

Symbol-free definition
The fully invariant closure of a subgroup in a group can be defined in any of the following equivalent ways:


 * As the intersection of all defining ingredient::fully invariant subgroups containing the given subgroup
 * As the subgroup generated by all defining ingredient::endomorphs to the given subgroup
 * As the set of all elements that can be written as products of finite length of elements from the subgroup and their endomorphic images

Definition with symbols
The fully invariant closure of a subgroup $$H$$ in a group $$G$$, is defined in the following equivalent ways:


 * As the intersection of all fully invariant subgroups of $$G$$ containing $$H$$
 * As the subgroup generated by all $$\rho(H)$$ where $$\rho \in \operatorname{End}(G)$$

Related operators

 * Characteristic closure
 * Normal closure