# Left residual operator for composition

## Contents

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This is a binary subgroup property operator, viz an operator that takes as input two subgroup properties, and outputs one subgroup property

## Definition

### Property-theoretic statement

Suppose $p,q$ are two subgroup properties. The left residual of $p$ by $q$ is the unique subgroup property $r$ such that:

$s * q \le p \iff s \le r$.

Here, $*$ denotes the composition operator.

### Statement with symbols

Suppose $p,q$ are two subgroup properties. The left residual of $p$ by $q$ is defined as the subgroup property $r$ as follows:

$H$ has property $r$ in $G$ if whenever $G$ is embedded as a subgroup with property $q$ in a group $K$, $H$ has property $p$ in $K$.

## Facts

### Relation with left transiter

Further information: Left transiter

The left transiter of a subgroup property $p$ is defined as the left residual of $p$ by itself.