Full invariance is strongly join-closed

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

Suppose $G$ is a group, and $H_{i},i\in I$ are fully invariant subgroups of $G$. Then, the join of subgroups $\langle H_{i}\rangle$ is also a fully invariant subgroup.

Related facts

A generalization is:

endo-invariance implies strongly join-closed

Other instances of the generalization are:

• Normality is strongly join-closed
• Characteristicity is strongly join-closed
• Strict characteristicity is strongly join-closed

Facts used

1. Endo-invariance implies strongly join-closed

Proof

The proof follows directly from fact (1), since full invariance is an endo-invariance property.