# Upward-closure 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)

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

## Definition

### Symbol-free definition

Let be a subgroup property. Then the upward closure of is defined as the property of being a subgroup such that all subgroups containing it satisfy property in the whole group.

### Definition with symbols

Let be a subgroup property. Then, the upward closure of is defined as the following subgroup property : A subgroup satisfies property in , if for every subgroup with , satisfies in .

## Properties

Template:Idempotent subgroup property modifier

Applying the upward closure operator twice is the same as applying it once. In other words, the properties that are *fixed* under the upward closure operator are precisely the same as the properties that can be obtained as images of the upward closure operator. A property that is fixed under the upward closure operator is termed an upward-closed subgroup property.

Template:Monotone subgroup property modifier

If (both are subgroup properties) then the where denotes the upward closure.