Normal-potentially characteristic subgroup

Definition
A subgroup $$H$$ of a group $$K$$ is termed normal-potentially characteristic in $$K$$ if there exists a group $$G$$ containing $$K$$ such that:


 * $$K$$ is a defining ingredient::normal subgroup of $$G$$.
 * $$H$$ is a defining ingredient::characteristic subgroup of $$G$$.

In terms of the upper-hook operator
Given two subgroup properties $$p$$ and $$q$$, the upper-hook operator of $$p$$ and $$q$$ is defined as the following property $$r$$: a subgroup $$H$$ of a group $$K$$ has property $$r$$ if there exists a group $$G$$ containing $$K$$ such that $$H$$ has property $$p$$ in $$G$$ and $$K$$ has property $$q$$ in $$G$$.

The property of being semi-strongly potentially characteristic is thus obtained by applying the upper-hook operator to the properties characteristic subgroup and normal subgroup.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Characteristic-potentially characteristic subgroup

Weaker properties

 * Stronger than::Normal-potentially relatively characteristic subgroup
 * Stronger than::Normal-extensible automorphism-invariant subgroup
 * Stronger than::Normal subgroup