On affine difference sets and their multipliers (Q1025469)

Let \(D\) be an affine difference set of order \(n\) in an abelian group \(G\) relative to a subgroup \(N\). We denote by \(\pi(s)\) the set of primes dividing an integer \(s(>0)\) and set \(H^*= H\setminus\{\omega\}\), where \(H=G/N\) and \(\omega= \prod_{\sigma\in H}\sigma\). In this article, using \(D\) we define a map \(g\) from \(H\) to \(N\) satisfying for \(\tau,\rho\in H^*\), \(g(\tau)= g(\rho)\) iff \(\{\tau,\tau-1\}= \{\rho,\rho-1\}\) and show that \(\text{ord}_{0(\sigma)}(m)/ \text{ord}_{0(g(\sigma))}(m)\in \{1,2\}\) for any \(\sigma\in H^*\) and any integer \(m>0\) with \(\pi(m)\subset\pi(n)\). This result is a generalization of \textit{J. C. Galati}'s theorem on even order \(n\) [Discrete Math. 306, No.~1, 42--51 (2006; Zbl 1085.05020)] and gives a new proof of a result of Arasu-Pott on the order of a multiplier modulo \(\exp(H)\) [\textit{K. T. Arasu} and \textit{A. Pott}, Des. Codes Cryptography 1, No.~1, 83--92 (1991; Zbl 0739.51009), Section 5].