# Normal-potentially characteristic subgroup

From Groupprops

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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

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## Definition

A subgroup of a group is termed **normal-potentially characteristic** in if there exists a group containing such that:

- is a normal subgroup of .
- is a characteristic subgroup of .

## 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 and , the upper-hook operator of and is defined as the following property : a subgroup of a group has property if there exists a group containing such that has property in and has property in .

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.