Characteristicity is strongly join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., strongly join-closed subgroup property)
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Statement

Statement with symbols

Suppose $G$ is a group and $H_{i},i\in I$ is a (possibly empty) collection of characteristic subgroups of $G$ . Suppose the join of the $H_{i}$ s equals $H$ . By convention, the join of the empty collection is taken to be the trivial subgroup.

Then, $H$ is also a characteristic subgroup of $G$ .

Related facts

Related facts about characteristicity

Generalizations

The statement has a generalization that states that any endo-invariance property is strongly join-closed. Here, endo-invarance means the proprty of being invariant under endomorphisms satisfying some given property. This fact, in turn, follows from the fact that homomorphisms commute with joins.

Other instances of the generalization are:

Property	Endo-invariance property with respect to ...	Proof that it is strongly join-closed
Normal subgroup	inner automorphisms	Normality is strongly join-closed
Fully invariant subgroup	endomorphisms	Full invariance is strongly join-closed
Strictly characteristic subgroup	surjective endomorphisms	Strict characteristicity is strongly join-closed
Injective endomorphism-invariant subgroup	injective endomorphisms	Injective endomorphism-invariance is strongly join-closed

Join-closedness for related properties

Proof

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