Cyclotomic Fields in Finite Geometry 2025, 36 - Field Descent Exponent Bounds

For a (n, k, λ, n)-difference set D in a group G, U a normal subgroup of G with G/U cyclic of order e, then

If G is abelian, then the same thing applies to exp G, and if G is cyclic, then instead

For finite set P of primes there is a computable constant C(P) with expG ≤ C(P)|G|^1/2 for abelian G containing a Hadamard difference set with order u² a product of powers of primes in P. If there is a Hadamard difference set in a cyclic group with order v = 4u², then F(v, u)²/φ(F(v, u)) ≥ v. For McFarland parameters in a cyclic group of order q^d+1[(q^d+1 - 1)/(q - 1) + 1], q = p^f, then p > 2, d = f = 1, (p+2)/(φ(p+2)) ≥ 4 - 12/(p+2). p+2 has at least 20 distinct prime divisors, and p > 2×10²⁸. There are no difference sets with Spence, CDJ parameters in cyclic groups. In application, no circulant Hadamrd matrix of order v in range 4 < v ≤ 10¹¹ with possible exceptions of v = 4u², u ∈ {165, 11715, 82005}.

For a (m, n, k, λ)-difference R set in a group G relative to N, U a subgroup of G without N, so that G/U is cyclic and has order e, then

A collineation group G is "quasiregular" if inducing regular operations on all point orbits. There is a projective plane with ord n with quasilinear G of ord n² and point orbits of size 1, n, n² iff there is an (n, n, n, 1)-difference set R in G relative to a normal subgroup N. If p is a prime divisor of n and S the Sylow p-subgroup of N, if p^a is the power of p dividing n, then exp S ≤ p^μ where μ is the smallest integer ≥ a/2.

Any finite set P of primes have computable constant C(P) with exp H, exp N ≤ C(P) √n. for any abelian groups H, N and planar function f: H → N with degree n, being a product of powers of primes in P. With a prime power q and d >_ℕ 3, define a circulant weighing matrix W((q^d - 1)/(q - 1), q^d-1). It exists iff d is odd. For a set of finite primes P and the set of all products of powers of primes in P Q, then there is a computable constant C(P) with exp(G) ≤ C(P)s for all s ∈ Q and any abelian group G of order 2s² with some G-invariant weighting matrix W. It can only exist for finitely many s ∈ Q.