Normal-potentially characteristic subgroup

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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]

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This term is related to: NPC conjecture
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Definition

A subgroup $H$ of a group $K$ is termed semi-strongly 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

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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.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Strongly potentially characteristic subgroup

Weaker properties

• Potentially characteristic subgroup: For proof of the implication, refer Semi-strongly potentially characteristic implies potentially characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Potentially characteristic not implies semi-strongly potentially characteristic.
• Semi-strongly potentially relatively characteristic subgroup
• Normal subgroup: For proof of the implication, refer Semi-strongly potentially characteristic implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies semi-strongly potentially characteristic.