# Subordination operator

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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)

## Contents

## Definition

### Symbol-free definition

The subordination operator is a map from the subgroup property space to itself that sends a subgroup property to the property of being a subgroup for which there exists a ascending chain of subgroups from the subgroup to the group with each member satisfying in its successor.

### Definition with symbols

The subordination operator on a property gives the following property: satisfies it in if there is an ascending chain ≤ ≤ with each satisfying in .

## Property theory of the subordination operator

### Transitive and identity-true

As for a general Kleene star operator, the subordination operator is a monotone descendant operator and is also idempotent. The fixed points are precisely the transitive identity-true properties.