Periodic normal implies image-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., image-potentially characteristic subgroup)
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This fact is related to: NIPC conjecture
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Statement

Suppose $H$ is a periodic normal subgroup of a group $G$. Then, $H$ is an image-potentially characteristic subgroup of $G$.

Related facts

• Finite normal implies image-potentially characteristic, because finite normal implies amalgam-characteristic
• Central implies image-potentially characteristic, because central implies amalgam-characteristic
• Normal subgroup contained in hypercenter is image-potentially characteristic, because normal subgroup contained in hypercenter is amalgam-characteristic
• Periodic normal implies potentially characteristic

Facts used

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

Proof

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