Fully invariant implies finite direct power-closed characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., fully invariant subgroup) must also satisfy the second subgroup property (i.e., finite direct power-closed characteristic subgroup)
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Statement

Any fully invariant subgroup of a group is a finite direct power-closed characteristic subgroup.

Related facts

Converse

The converse is not true: finite direct power-closed not implies fully invariant.

Other related facts

• Bound-word implies finite direct power-closed characteristic
• Full invariance is finite direct power-closed
• Characteristicity is not finite direct power-closed

Facts used

1. Fully invariant implies characteristic
2. Full invariance is finite direct power-closed

Proof

The proof follows directly from facts (1) and (2).