Fully invariant closure

This article defines a subgroup operator related to the subgroup property full characteristicity. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

Symbol-free definition

The fully characteristic closure of a subgroup in a group can be defined in any of the following equivalent ways:

As the intersection of all fully characteristic subgroups containing the given subgroup
As the subgroup generated by all 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 characteristic closure of a subgroup $H$ in a group $G$ , is defined in the following equivalent ways:

As the intersection of all fully characteristic subgroups of $G$ containing $H$
As the subgroup generated by all $\rho (H)$ where $\rho \in End(G)$

Relation with other operators

Related operators