# Right transiter

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 property

View 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:t.i. subgroup property

## Definition

### Symbol-free definition

The right transiter is an operator from the subgroup property space to itself, defined as follows. The right transiter of a subgroup property $p$ is the maximum among all subgroup properties $q$ for which the following holds:

$p * q \le p$

where $*$ denotes the composition operator on subgroup properties.

### Definition with symbols

The right transiter of a subgroup property $p$ is the property $q$ defined by the following criterion:

A subgroup $H$ in a group $G$ is said to have $q$ if and only if whenever $K$ has property $p$ as a subgroup of $H$, $K$ also has property $p$ in $G$.

We denote the right transiter of a property $p$ as $R(p)$.

### In terms of the residual operator

The right transiter of a property is its right residual by itself, with respect to the composition operator.

A general notion of right transiter can be given for any associative quantalic binary operator on a property space. The generic notion inherits many of the nice behaviours from this special case.

## Effect on subgroup metaproperties

If $p$ satisfies the intermediate subgroup condition, so does the right transiter of $p$.

## Computing the right transiter

### In the function restriction formalism

If a subgroup property can be expressed by a function restriction formal expression $a \to b$, thne we can do the following: