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Hadamard difference sets in groups with a cyclic odd quotient

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Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by an independent AI review and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

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Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 04, 2026