Finite-p-potentially characteristic subgroup

Statement

Suppose $p$ is a prime number and $G$ is a finite $p$-group (i.e., a group of prime power order). In other words, $G$ is a group of prime power order. A subgroup $H$ of $G$ is termed a finite-$p$-potentially characteristic subgroup if there exists a finite $p$-group $K$ containing $G$ such that $H$ is a characteristic subgroup of $K$.

Facts

• Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it

Relation with other properties

The generalization of this property to finite groups, rather than just finite $p$-groups, is the property of being a finite-pi-potentially characteristic subgroup.

Stronger properties

• Characteristic subgroup of group of prime power order
• Central subgroup of group of prime power order
• Cyclic normal subgroup of group of prime power order
• Homocyclic normal subgroup of group of prime power order

Weaker properties

• Normal subgroup of group of prime power order