# Intermediately 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)

*This property modifier is idempotent and a property is a fixed-point, or equivalently, an image of this if and only if it is a:*intermediate subgroup condition

## Contents

## Definition

### Symbol-free definition

The *intermediately* operator is a map from the subgroup property space to itself, that sends a subgroup property to the property of being a subgroup that satisfies not only in the whole group, but also in every intermediate subgroup.

### Definition with symbols

Given a subgroup property , the subgroup property *intermediately* is the property as follows: satisfies *intermediately* in if for any group with , satisfies in .

## Properties

### Monotonicity

*This subgroup property modifier is monotone, viz if are subgroup properties and is the operator, then *

If are two subgroup properties, then *intermediately* *intermediately* . This follows directly from the definition.

### 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 , *intermediately* . This follows from the fact that if satisfies property in *every* intermediate subgroup, also satisfies property in the *whole* group.

### Idempotence

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

The intermediately operator is idempotent, in the sense that applying the intermedaitely operator twice has the same effect as applying it once. The image-cum-fixed-point-space for this operator is precisely the subgroup properties satisfying the intermediate subgroup condition.

## Effect on metaproperties

Template:Join-closedness-preserving

Suppose is a join-closed subgroup property, viz the join of any family of subgroups satisfying property , also satisfies property . Then, it is easy to see that *intermediately* is also join-closed.

### Transitivity

It is not clear whether, even if is transitive, *intermediately* will b transitive.

### Transfer condition

If satisfies the transfer condition, it also, in particular, satisfies the intermediate subgroup condition, and hence is unchanged under application of the intermediately operator.

## Properties obtained via this operator

### Naturally arising properties that satisfy intermediate subgroup condition

These include: