Extraneous Multipliers of Abelian Difference Sets
Wandi Wei, Shuhong Gao, Qing Xiang · 2020
Let (G,·) be a group of order v. A k-subset D of G is called a (v,k, λ ) difference set if the list of differences d 1 d 2 − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071907/8a7013d4-661b-41a7-ba62-74d724e48458/content/eq1689.tif"/> , d 1 ,d 2 ∈ G, contains each non-identity element of G exactly λ times. The difference set D is said to be abelian (resp. cyclic) if G is abelian (resp. cyclic). The number n = k − λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071907/8a7013d4-661b-41a7-ba62-74d724e48458/content/eq1690.tif"/> is called the order of the difference set. An automorphism α of G is called a multiplier of D if D α = aDb https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071907/8a7013d4-661b-41a7-ba62-74d724e48458/content/eq1691.tif"/> for some a,b ∈ G. When a is the identity element, α is called a right multiplier. Let G be abelian. We know that for any positive integer t relatively prime to v, the mapping x ↦ x t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071907/8a7013d4-661b-41a7-ba62-74d724e48458/content/eq1692.tif"/> is an automorphism of G. If it happens to be a multiplier of D, then it is called a numerical multiplier of D. In this case we sometimes say that t is a multiplier by abuse of terminology. The following theorem suggests that the factors of n are an 166 important source of multipliers of an abelian difference set of order n.