Group property-conditionally potentially fully invariant subgroup

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Definition

Suppose $\alpha$ is a group property, $G$ is a group satisfying $\alpha$, and $H$ is a subgroup of $G$. We say that $H$ is a potentially fully invariant subgroup of $G$ conditional to $\alpha$ (or relative to $\alpha$) if there exists a group $K$ containing $G$ and satisfying $\alpha$ such that $H$ is a fully invariant subgroup of $K$.

Particular cases

Group property	Property of being a potentially fully invariant subgroup
any group	potentially fully invariant subgroup
finite group	finite-potentially fully invariant subgroup
group of prime power order	finite-p-potentially fully invariant subgroup
finite group with a given set of prime factors	finite-pi-potentially fully invariant subgroup

Relation with other properties

Stronger properties

• Group property-conditionally potentially verbal subgroup

Weaker properties

• Group property-conditionally potentially characteristic subgroup