Characteristicity is strongly intersection-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about strongly intersection-closed subgroup property

Statement

Verbal statement

An arbitrary (possibly empty) intersection of characteristic subgroups of a group is a characteristic subgroup.

Statement with symbols

Suppose $I$ is an indexing set and $H_{i}, i \in I$ is a collection of characteristic subgroups of a group $G$ . Then, the intersection of subgroups $⋂_{i \in I} H_{i}$ is also a characteristic subgroup of $G$ .

Related facts

Generalizations

Invariance implies strongly intersection-closed: Any invariance property (i.e., a property that can be expressed as invariance under a certain collection of functions) is strongly intersection-closed: an arbitrary intersection of subgroups with the property again has the property.

Other particular cases of this general result are:

Analogues in other algebraic structures

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

Proof

Hands-on proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Property-theoretic proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]