Relatively characteristic subgroup

This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
View other such properties

Definition

Definition with symbols

Let $H\leq K\leq G$ be groups. Then we say that $H$ is relatively characteristic in $K$ with respect to $G$ if any automorphism of $G$ that restricts to an automorphism of $K$ must in fact restrict to an automorphism of $H$.

Relation with other properties

Stronger properties

• Middle-characteristic subgroup: If $H\leq K\leq G$ and $H$ is characteristic in $K$, it is relatively characteristic in $K$ with respect to $G$.
• Characteristic subgroup: If $H\leq K\leq G$ and $H$ is characteristic in $G$, it is relatively characteristic in $K$ with respect to $G$.

Weaker properties

• Normalizer-relatively normal subgroup
• Relatively normal subgroup

A related subgroup property

A subgroup $H$ of a group $K$ is said to be potentially relatively characteristic in $K$ if there exists a group $G$ containing $K$ such that $H$ is relatively characteristic in $K$, relative to $G$. Potentially relatively characteristic subgroups are precisely the same as normal subgroups.