{"id":"hexagon/2611.00102","pdf":"https://hexagonmath.org/pdf/2611.00102v1","url":"https://hexagonmath.org/abs/2611.00102v1","tags":["math.NT","msc:05B10","msc:05B20","msc:11R18","Hadamard difference sets","Lander's conjecture","Menon difference sets","circulant Hadamard matrices","cyclotomic integers","group rings"],"score":0,"title":"Hadamard difference sets in groups with a cyclic odd quotient","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}