Characteristic-potentially characteristic subgroup

Symbol-free definition
A subgroup of a group is termed characteristic-potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed characteristic-potentially characteristic in $$G$$ if there exists a group $$K$$ containing $$G$$ such that both $$H$$ and $$G$$ are characteristic in $$K$$.

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 strongly potentially characteristic is thus obtained by applying the upper-hook operator to the property characteristic subgroup with itself.

Stronger properties

 * Weaker than::Characteristic subgroup

Weaker properties

 * Stronger than::Normal-potentially characteristic subgroup
 * Stronger than::Normal-potentially relatively characteristic subgroup
 * Stronger than::Potentially characteristic subgroup:
 * Stronger than::Normal subgroup:
 * Potentially relatively characteristic subgroup: This is a weaker notion, that turns out to be the same as normality.
 * Stronger than::Characteristic-extensible automorphism-invariant subgroup
 * Stronger than::Normal-extensible automorphism-invariant subgroup