Periodic normal implies potentially 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., periodic normal subgroup) must also satisfy the second subgroup property (i.e., potentially characteristic subgroup)
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Statement

A periodic normal subgroup of a group is a potentially characteristic subgroup.

Related facts

Similar facts

• Finite normal implies potentially characteristic, uses finite normal implies amalgam-characteristic.
• Central implies potentially characteristic, uses central implies amalgam-characteristic.
• Normal subgroup contained in hypercenter is potentially characteristic, uses normal subgroup contained in hypercenter is amalgam-characteristic.
• Abelian implies every subgroup is potentially characteristic
• Nilpotent implies every normal subgroup is potentially characteristic
• Group with no element of order p implies every p-divisible normal subgroup is potentially characteristic

Facts used

1. Periodic normal implies amalgam-characteristic
2. Amalgam-characteristic implies potentially characteristic

Proof

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