Difference set

This term is related to: incidence geometry
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This article defines a property of subsets of groups
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Definition

A subset $D$ of a finite group $G$ is termed a difference set if every non-identity element of $G$ can be written as a right quotient of elements of $D$ in exactly $\lambda$ ways, for some fixed $\lambda$ 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 $\lambda$ where $\lambda$ is the number of ways of writing each non-identity element as a right quotient of elements of the difference set.

Related notions

• Relative difference set for a subgroup
• Divisible difference set for a subgroup

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.