# Maximal among abelian subgroups

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

This article describes a property that arises as the conjunction of a subgroup property: self-centralizing subgroup with a group property (itself viewed as a subgroup property): Abelian group

View a complete list of such conjunctions

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed **maximal among abelian subgroups** or a **maximal abelian subgroup** or a **self-centralizing abelian subgroup** if it satisfies the following equivalent conditions:

- It equals its centralizer in the whole group.
- It is abelian and self-centralizing.
- It is abelian and is not properly contained in a bigger abelian subgroup.

### Definition with symbols

A subgroup of a group is termed **maximal among Abelian subgroups** or a **maximal abelian subgroup** or a **self-centralizing abelian subgroup** if it satisfies the following equivalent conditions:

- , where denotes the centralizer of in .
- and is abelian.
- is abelian, and if with Abelian, then .

### Equivalence of definitions

`For full proof, refer: Equivalence of definitions of maximal among abelian subgroups`

## Formalisms

### In terms of the maximal operator

This property is obtained by applying the maximal operator to the property: Abelian subgroup

View other properties obtained by applying the maximal operator

## Relation with other properties

### Stronger properties

- In a nilpotent group or supersolvable group, maximal among abelian normal subgroups:
`Further information: Maximal among abelian normal implies self-centralizing in nilpotent, maximal among abelian normal implies self-centralizing in supersolvable`

### Weaker properties

- c-closed self-centralizing subgroup
- c-closed subgroup
- Self-centralizing subgroup
- Subgroup containing the center

## Facts

- In any group, there always exist maximal among abelian subgroups. In fact, every abelian subgroup of a group is contained in a maximal among abelian subgroup.
`Further information: Every abelian subgroup is contained in a maximal among abelian subgroups` - Two subgroups that are maximal among abelian subgroups need not be isomorphic. In fact, they may not even have the same size. For instance, in the symmetric group on three letters, there is a subgroup of order two and a subgroup of order three, both of them maximal among abelian subgroups.
- In fact, any group can be expressed as a union of subgroups that are maximal among abelian subgroups. In particular, any non-abelian group has at least three distinct maximal among abelian subgroups.
`Further information: Every group is a union of maximal among abelian subgroups` - In certain cases, any abelian subgroup can be replaced by a normal subgroup or 2-subnormal subgroup of the same size.
`Further information: Category:Replacement theorems`