On the Multiplier Conjecture

The Multiplier Theorem is a celebrated theorem in the Design theory.The condition p】λis crucial to all known proofs of the multiplier theorem.However in all known examples of difference sets μp is a multiplier for every prime p with（p,v）=1 and pln.Thus there is the multiplier conjecture:"The multiplier theorem holds without the assumption that p】 λ."The general form of the multiplier theorem may be viewed as an attempt to partially resolve the multiplier conjecture,where the assumption"p】λ"is replaced by "n1】λ".Since then Newman（1963）,Turyn（1964）,and McFarland（1970）attempted to partially resolve the multiplier conjecture（see[7],[8],[9]）.This paper will prove the following result using the representation theory of finite groups and the algebraic number theory:Let G be an abelian group of order v,v0 be the exponent of G,and D be a（v, k,λ）-difference set in G.If n=2n1, then the general form of the multiplier theorem holds without the assumption that n1】λin any of the following cases: （1）2|n1; （2）2 ×n1,and（v,7）=1; （3） 2 ×n1,7|v,and t≡1 or 2 or 4（mod 7）.