# Maximal among abelian normal subgroups

From Groupprops

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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]

## Definition

### Symbol-free definition

A subgroup of a group is termed **maximal among Abelian normal subgroups** if it is an Abelian normal subgroup and there is no Abelian normal subgroup properly containing it.

### Definition with symbols

A subgroup of a group is termed **maximal among Abelian normal subgroups** if is an Abelian normal subgroup of , and for any containing that is an Abelian normal subgroup of , .

## Formalisms

### In terms of the maximal operator

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

View other properties obtained by applying the maximal operator

## Relation with other properties

### Weaker properties

### Related properties

- Maximal among Abelian characteristic subgroups
- Self-centralizing subgroup (if inside a supersolvable group):
`For full proof, refer: Maximal among Abelian normal implies self-centralizing in supersolvable`