Necessary conditions for Menon difference sets (Q1802194)

A \((v,k,\lambda)\)-difference set \(D\) in a group \(G\) is a \(k\)-subset of \(G\) such that the list of differences \(d-d'\) with distinct elements \(d\), \(d'\in D\) contains each nonzero group element exactly \(\lambda\) times. One of the most interesting families of difference sets is the series of Menon-Hadarmard difference sets which have parameters \((4n^ 2,2n^ 2- n,n^ 2-n)\) and correspond to group invariant Hadamard matrices. Many nonexistence results on difference sets in abelian groups (in particular on Menon-Hadamard difference sets) are known under the assumption that a prime divisor \(p\) of \(k-\lambda\) exists such that \(p^ j \equiv-1\) modulo the exponent of \(H\) (where \(H\) is a complement of the Sylow \(p\)- subgroup of \(G)\). This paper contains an interesting new approach to prove nonexistence results without this assumption. The author can show, for instance, that a group \(\mathbb{Z}_{6p} \times\mathbb{Z}_{6p}\) \((p\) prime) contains no Menon-Hadamard difference sets if \(p \neq 3\), 13 (no example is known if \(p=13)\).