# Abelian Sylow subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: Sylow subgroup with a group property (itself viewed as a subgroup property): abelian group

View a complete list of such conjunctions

## Definition

A subgroup of a group is termed an **abelian Sylow subgroup** if it is abelian as a group and is also a Sylow subgroup of the whole group.

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Sylow subgroup | |FULL LIST, MORE INFO | |||

abelian pronormal subgroup | abelian and a pronormal subgroup | |FULL LIST, MORE INFO | ||

SCDIN-subgroup (not standard terminology) |
any conjugation between two subsets in the subgroup via an element of the whole group can also be realized via an element in the normalizer of the subgroup | abelian and Sylow implies SCDIN | Abelian pronormal subgroup|FULL LIST, MORE INFO |