# Maximal operator

This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup propertyView a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

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## Definition

The **maximal operator** is a subgroup property modifier that takes as input a subgroup property and outputs a subgroup property defined as follows: has property in if the following three hold:

- has property in
- There is no subgroup of , containing , such that satisfies .

In other words, is a maximal element (in the containment partial order) among those proper subgroups of that satisfy .

## Application

Some important instances of application of the maximal operator:

- maximal subgroup: obtained from proper subgroup
- maximal normal subgroup: obtained from proper normal subgroup
- maximal characteristic subgroup: obtained from proper characteristic subgroup

For a complete list, refer:

Category:Subgroup properties obtained by applying the maximal operator

## Properties

### Monotonicity

The maximal operator is *not* monotone. In other words, if are subgroup properties, then a maximal -subgroup need not be a maximal -subgroup. The reason is that there may be bigger intermediate subgroups that satisfy property but not property .

### Descendance

*This subgroup property modifier is descendant, viz the image of any subgroup property under this modifier is stronger than that property. In symbols, if denotes the modifier and and property, *

For any subgroup property , the property of being a maximal -subgroup, is stronger than the property of being a -subgroup.

### Idempotence

*This subgroup property modifier is idempotent, viz applying it twice to a subgroup property has the same effect as applying it once*

Clearly, applying the maximal operator twice has the same effect as applying it once. Those subgroup properties that are obtained by applying this operator are termed maximal subgroup properties.