Characteristicity is strongly join-closed

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 about characteristicity

 * Characteristicity is strongly intersection-closed
 * Characteristicity does not satisfy intermediate subgroup condition
 * Characteristicity is not upper join-closed

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:

Join-closedness for related properties

 * Closure-characteristicity is strongly join-closed
 * Automorph-conjugacy is not finite-join-closed
 * Procharacteristicity is not finite-join-closed