Forbidden multipliers in abelian difference sets

TL;DR

This paper identifies specific automorphisms in abelian groups that cannot serve as multipliers in non-trivial difference sets, except for a special involution case in Hadamard difference sets, leading to new non-existence results.

Contribution

It introduces a novel criterion for ruling out certain automorphisms as multipliers in abelian difference sets and establishes bounds on their multiplier groups.

Findings

Certain automorphisms cannot be multipliers in non-trivial difference sets

The involution can be a multiplier in Hadamard difference sets under specific conditions

Bounds are derived for the numerical multiplier group of a difference set

Abstract

We make the observation that certain group automorphisms that fix a large subgroup of an abelian group cannot be multipliers in any non-trivial abelian difference sets, with the single exception of an involution that can be a multiplier in Hadamard difference sets, provided that the difference set contains a sub-difference set of the same type. We use this observation together with a multiplier theorem to rule out the existence of difference sets, and derive bounds for the numerical multiplier group of a difference set.