Difference set

In <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, a 
 
 
 
 (
 v
 ,
 k
 ,
 λ
 )
 
 
 {\displaystyle (v,k,\lambda )}
 
 difference set is a subset 
 
 
 
 D
 
 
 {\displaystyle D}
 
 of <a href="/facts/Cardinality/m6VPojwT">size</a> 
 
 
 
 k
 
 
 {\displaystyle k}
 
 of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 of <a href="/facts/Order_of_a_group/Cl3tKGFg">order</a> 
 
 
 
 v
 
 
 {\displaystyle v}
 
 such that every non-<a href="/facts/Identity_element/9nss3xHY">identity</a> element of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 can be expressed as a product 
 
 
 
 
 d
 
 1
 
 
 
 d
 
 2
 
 
 −
 1
 
 
 
 
 {\displaystyle d_{1}d_{2}^{-1}}
 
 of elements of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 in exactly 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 ways. A difference set 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is said to be cyclic, abelian, non-abelian, etc., if the group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 has the corresponding property. A difference set with 
 
 
 
 λ
 =
 1
 
 
 {\displaystyle \lambda =1}
 
 is sometimes called planar or simple. If 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> written in additive notation, the defining condition is that every non-zero element of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 can be written as a difference of elements of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 in exactly 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 ways. The term "difference set" arises in this way.

Difference set open-in-new

Difference set