Finite-pi-potentially fully invariant subgroup

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Definition

Let $K$ be a group, and $H$ be a subgroup of $K$. We say that $H$ is a finite-pi-potentially fully invariant subgroup of $K$ if there exists a finite group $G$ containing $K$ such that every prime factor of the order of $G$ also divides the order of $K$, and such that $H$ is a fully invariant 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
• Finite-pi-potentially verbal subgroup

Weaker properties

• Finite-pi-potentially characteristic subgroup