On difference sets in groups of order \(4p^ 2\)
The paper under review is concerned with Menon difference sets, i.e. difference sets \(D\) with parameters of the form \(v = 4u^2\), \(k = 2u^2 \pm u\) and \(\lambda = u^2 \pm u\) (also called ``Hadamard difference sets'' in the literature). These difference sets are of central interest; for background, see the survey by \textit{J. Davis} and \textit{J. Jedwab} [A survey of Hadamard difference sets, Groups, difference sets, and the monster, ed. \textit{K. T. Arasu} et al., 145-156 (1996)]. Up to now, most known results concern the abelian case. The author gives an important contribution to the general case (where the underlying group may be nonabelian) when \(u \geq 5\) is a prime: There are only two possible isomorphism types of groups if \(p \equiv 3 \pmod 4\) and six possible isomorphism types for \(p \equiv 1 \pmod 4\). Of course, all resulting groups are nonabelian by a well-known result of \textit{R. L. McFarland} [Difference sets in abelian groups of order \(4p^2\), Mitt. Math. Semin. Gießen 192, 1-70 (1989; Zbl 0695.05013)]. It should also be noted that a nonabelian example has been constructed for \(u = 5\) by \textit{K. W. Smith} [Non-abelian Hadamard difference sets, J. Comb. Theory, Ser. A 70, No. 1, 144-156 (1995; Zbl 0818.05015)].
scientific article; zbMATH DE number 823496
scientific article