# Minimal 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)

## Definition

The **minimal 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
- is a nontrivial subgroup of
- There is no nontrivial subgroup of , contained in , such that satisfies .

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

## Application

Some important instances of application of the maximal operator:

- minimal normal subgroup: obtained from normal subgroup
- minimal characteristic subgroup: obtained from characteristic subgroup

## Properties

### Monotonicity

The minimal 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 smaller 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 minimal -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 minimal subgroup properties.