# Group property-conditionally potentially characteristic subgroup

This term is related to: potentially characteristic subgroups characterization problem
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## Definition

Suppose $\alpha$ is a group property. Suppose $G$ is a group satisfying $\alpha$ and $H$ is a subgroup of $G$. We say that $H$ is potentially characteristic with respect to $\alpha$, or conditional to $\alpha$, if there exists a group $K$ containing $G$ such that $H$ is a characteristic subgroup of $K$.

## Particular cases

Group property Property of being a potentially characteristic subgroup
any group normal subgroup (proof: NPC theorem)
finite group normal subgroup of finite group (proof: Finite NPC theorem)
group of prime power order finite-p-potentially characteristic subgroup
group with fixed set of prime divisors finite-pi-potentially characteristic subgroup