# Fully invariant closure

From Groupprops

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.

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.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Fully invariant closure, all facts related to Fully invariant closure) |Survey articles about this | Survey articles about definitions built on this

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## Definition

### 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 fully invariant 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 invariant closure** of a subgroup in a group , is defined in the following equivalent ways:

- As the intersection of all fully invariant subgroups of containing
- As the subgroup generated by all where