# Maximal among abelian characteristic 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]

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is termed **maximal among Abelian characteristic subgroups** if it is an Abelian characteristic subgroup, and is not properly contained in any Abelian characteristic subgroup.

## Formalisms

### In terms of the maximal operator

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

View other properties obtained by applying the maximal operator

## Relation with other properties

### Weaker properties

- Center of critical subgroup for a group of prime power order
- Abelian characteristic subgroup
- Abelian normal subgroup

## Facts

### Facts about such subgroups in nilpotent groups and p-groups

- Maximal among Abelian characteristic not implies self-centralizing in nilpotent
- Thompson's critical subgroup theorem
- Maximal among Abelian characteristic not implies Abelian of maximum order

Here are some facts about the possibility of existence of multiple subgroups that are maximal among Abelian characteristic subgroups: