math.CO — Combinatorics

Hadamard difference sets in groups with a cyclic odd quotient

Contributed by David Niedbala Giraudin

Let u > 1 be odd. We prove that the abelian group C_2 x C_{2u^2} contains no Hadamard difference set. Together with the recently announced proof of the circulant Hadamard conjecture, this shows that an abelian group of order 4u^2 (u > 1 odd) whose odd Sylow subgroups are all cyclic never contains a Hadamard difference set, so Lander's conjecture holds for Hadamard difference sets in these groups. More generally, let G be a group of order 4u^2 with a normal subgroup N such that G/N is isomorphic to E x C_{w^2}, where |E| = 4, w > 1, w | u and gcd(w, u/w) = 1. If every prime divisor of u/w is self-conjugate modulo rad(w) (modulo 4 rad(w) when E is cyclic), then G contains no Hadamard difference set. The proof adapts the alternating character products used for the cyclic case.