Normal-potentially characteristic subgroup

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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 $G$.

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.
• 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.