Group property-conditionally potentially verbal 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 verbal subgroup conditional to $\alpha$ if there exists a group $K$ containing $G$ and satisfying $\alpha$ such that $H$ is a verbal subgroup of $K$.

Particular cases

Group property	Property of being a potentially verbal subgroup
any group	potentially verbal subgroup
finite group	finite-potentially verbal subgroup
finite group with fixed set of prime factors	finite-pi-potentially verbal subgroup

Relation with other properties

Weaker properties

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