Difference set

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

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.