# Abelian pronormal subgroup

From Groupprops

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

## Contents

## Definition

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

- is an abelian group.
- is a pronormal subgroup of : If is a conjugate of in , then are also conjugate in .

## 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 |