# Difference set

From Groupprops

This term is related to: incidence geometry

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This article defines a property of subsets of groups

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## Contents

## Definition

A subset of a finite group is termed a **difference set** if every non-identity element of can be written as a right quotient of elements of in exactly ways, for some fixed independent of the choice of element.

## Terminology for difference sets

The **order** of a difference set is defined as the cardinality of the difference set, minus where is the number of ways of writing each non-identity element as a right quotient of elements of the difference set.

## Related notions

## Notions of equivalence

### Isomorphic difference sets

Two difference sets (in possibly different groups) ar esaid to be **isomorphic** if they define isomorphic designs.

### Equivalent difference sets

Two difference sets in a group are said to be equivalent if there is an automorphism of the group that maps one difference set to a translate of the other.