# Difference between revisions of "Abelian pronormal subgroup"

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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 Center of 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) Center of pronormal subgroup|FULL LIST, MORE INFO
CDIN-subgroup conjugacy-determined subgroup in its normalizer (via SCDIN) SCDIN-subgroup|FULL LIST, MORE INFO
pronormal subgroup (by definition) |FULL LIST, MORE INFO