# Central subgroup of normalizer

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: central factor of normalizer 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 a **central subgroup of normalizer** if it satisfies the following equivalent conditions:

- It is a central subgroup (i.e., is contained in the center) of its normalizer.
- It is Abelian and is a central factor of normalizer.
- Any inner automorphism of the whole group that leaves the subgroup invariant, must act trivially on the subgroup.

## Formalisms

### In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: central subgroup

View other properties obtained by applying the in-normalizer operator

## Facts

- Burnside's normal p-complement theorem: This states that if a Sylow subgroup is a central subgroup of its normalizer, then it is a retract, i.e., it possesses a normal complement.

## Metaproperties

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: |ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition