Difference set

In combinatorics, a ${\displaystyle (v,k,\lambda )}$ difference set is a subset ${\displaystyle D}$ of size ${\displaystyle k}$ of a group ${\displaystyle G}$ of order ${\displaystyle v}$ such that every non-identity element of ${\displaystyle G}$ can be expressed as a product ${\displaystyle d_{1}d_{2}^{-1}}$ of elements of ${\displaystyle D}$ in exactly ${\displaystyle \lambda }$ ways. A difference set ${\displaystyle D}$ is said to be cyclic, abelian, non-abelian, etc., if the group ${\displaystyle G}$ has the corresponding property. A difference set with ${\displaystyle \lambda =1}$ is sometimes called planar or simple.^[1] If ${\displaystyle G}$ is an abelian group written in additive notation, the defining condition is that every non-zero element of ${\displaystyle G}$ can be written as a difference of elements of ${\displaystyle D}$ in exactly ${\displaystyle \lambda }$ ways. The term "difference set" arises in this way.