Finite-pi-potentially characteristic subgroup

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Definition

Let $K$ be a finite group and $H$ be a subgroup of $K$ . We say that $H$ is finite-pi-potentially characteristic in $K$ if there exists a finite group $G$ containing $K$ , such that every prime factor of $G$ divides $K$ , and $H$ is a characteristic subgroup of $G$ .

When $K$ is a group of prime power order, this is termed finite-p-potentially characteristic subgroup.

Relation with other properties

Stronger properties

Weaker properties

Normal subgroup of finite group