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