Group property-conditionally normal-potentially characteristic subgroup

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This term is related to: potentially characteristic subgroups characterization problem
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Definition

Let $H$ be a subgroup of a group $G$, and $\alpha$ be a group property. We say that $H$ is normal-potentially characteristic in $G$ relative to $\alpha$ if there exists a group $K$ containing $G$ and satisfying $\alpha$ such that $G$ is a normal subgroup of $K$ and $H$ is a characteristic subgroup of $K$.

If we simply use the term normal-potentially characteristic subgroup, it means that $\alpha$ is taken to be the tautology -- the property satisfied by all groups.

Relation with other properties

Stronger properties

• Group property-conditionally characteristic-potentially characteristic subgroup

Weaker properties

• Group property-conditionally potentially characteristic subgroup