Central 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., central subgroup) must also satisfy the second subgroup property (i.e., potentially characteristic subgroup)
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This fact is related to: NPC conjecture
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Statement

A central subgroup of a group is always a potentially characteristic subgroup: it can be realized as a characteristic subgroup inside a possibly bigger group.

Definitions used

Central subgroup

Further information: Central subgroup

A subgroup $H$ of a group $G$ is termed a central subgroup if every element of $H$ commutes with every element of $G$, or equivalently, if $H$ is contained in the center of $G$.

Potentially characteristic subgroup

Further information: Potentially characteristic subgroup

A subgroup $H$ of a group $G$ is termed a potentially characteristic subgroup if there exists a group $K$ containing $G$, such that, with the induced embedding, $H$ is a characteristic subgroup of $K$.

Related facts

Similar facts

• Finite NPC theorem, which states that any normal subgroup of a finite group is characteristic in some finite group containing it.
• Finite normal implies potentially characteristic, which uses the fact that finite normal implies amalgam-characteristic.
• Periodic normal implies potentially characteristic, which uses the fact that periodic normal implies amalgam-characteristic
• Normal subgroup contained in hypercenter is potentially characteristic, which uses the fact that normal subgroup contained in hypercenter is amalgam-characteristic
• Abelian implies every subgroup is potentially characteristic
• Nilpotent implies every normal subgroup is potentially characteristic
• Central implies potentially verbal in finite

Facts used

1. Central implies amalgam-characteristic
2. Amalgam-characteristic implies potentially characteristic

Proof

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