Group property-conditionally potentially characteristic subgroup

This term is related to: potentially characteristic subgroups characterization problem
View other terms related to potentially characteristic subgroups characterization problem | View facts related to potentially characteristic subgroups characterization problem

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

Relation with other properties

Stronger properties

• Group property-conditionally potentially verbal subgroup
• Group property-conditionally potentially fully invariant subgroup
• Group property-conditionally normal-potentially characteristic subgroup
• Group property-conditionally characteristic-potentially characteristic subgroup