# Normal-potentially characteristic subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This term is related to: potentially characteristic subgroups characterization problem
View other terms related to potentially characteristic subgroups characterization problem | View facts related to potentially characteristic subgroups characterization problem

## 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 normal subgroup of $G$.
• $H$ is a characteristic subgroup of $G$.

## Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### 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.