Characteristicity is strongly join-closed

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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_is 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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