Abelian pronormal subgroup

This article describes a property that arises as the conjunction of a subgroup property: pronormal subgroup with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions

Definition

A subgroup $H$ of a group $G$ is termed an abelian pronormal subgroup of $G$ if it satisfies the following two conditions:

1. $H$ is an abelian group.
2. $H$ is a pronormal subgroup of $G$: If $K$ is a conjugate of $H$ in $G$, then $H,K$ are also conjugate in $\langle H,K\rangle$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
central subgroup	contained in the center			|FULL LIST, MORE INFO
abelian normal subgroup	both abelian as a group and a normal subgroup			|FULL LIST, MORE INFO
abelian Sylow subgroup	both abelian as a group and a Sylow subgroup			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
center of pronormal subgroup	center of a pronormal subgroup	any abelian pronormal subgroup is its own center		|FULL LIST, MORE INFO
central subgroup of pronormal subgroup	central subgroup of a pronormal subgroup			|FULL LIST, MORE INFO
SCDIN-subgroup	subset-conjugacy-determined subgroup in its normalizer	abelian and pronormal implies SCDIN (also, via center of pronormal subgroup)		|FULL LIST, MORE INFO
CDIN-subgroup	conjugacy-determined subgroup in its normalizer	(via SCDIN)		|FULL LIST, MORE INFO
pronormal subgroup		(by definition)		|FULL LIST, MORE INFO