Finite-pi-potentially verbal subgroup

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Definition

Suppose $K$ is a finite group and $H$ is a subgroup of $K$. We say that $H$ is a finite-pi-potentially verbal subgroup of $K$ if the following holds: There exists a group $G$ containing $K$ such that all prime factors of the order of $G$ also divide the order of $K$ and $H$ is a verbal subgroup of $G$.

Relation with other properties

Stronger properties

• Central subgroup of finite group
• Cyclic normal subgroup of finite group
• Homocyclic normal subgroup of finite group

Weaker properties

• Finite-potentially verbal subgroup
• Finite-pi-potentially characteristic subgroup