Conjugation-invariantly relatively normal subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Conjugation-invariantly relatively normal subgroup, all facts related to Conjugation-invariantly relatively normal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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

Suppose $H\leq K\leq G$. We say that $H$ is conjugation-invariantly relatively normal in $K$ with respect to $G$ if $H$ is a normal subgroup in every conjugate of $K$ in $G$ that contains $H$.

Relation with other properties

Stronger properties

• Strongly closed subgroup
• Weakly closed subgroup: For proof of the implication, refer Weakly closed implies conjugation-invariantly relatively normal and for proof of its strictness (i.e. the reverse implication being false) refer Conjugation-invariantly relatively normal not implies weakly closed.

Weaker properties

• Relatively normal subgroup