Sharp Asymptotics for Abelian Covers of Groups with Bounded Noncommutativity

Abstract

We determine the sharp exponential growth rate of the minimum number of abelian subgroups required to cover a group with bounded pairwise noncommutativity. Let omega(G) denote the largest size of a pairwise noncommuting subset of a group G, let a(G) be the least size of an abelian cover, and define h(n) as the supremum of a(G) over groups satisfying omega(G) ≤ n. Answering a quantitative question posed by Erdős, we prove that log₂ h(n) = n/2 + O(sqrt(n)(log(n+2))³), and hence that h(n)^(1/n) tends to sqrt(2). Extraspecial 2-groups provide the matching lower bound at the exponential scale. The upper bound combines a central-factor analysis of finite p-groups with alternating-form clique estimates, interaction control across central factors, Sylow decomposition, and a polynomial-cost reduction to the Fitting subgroup. The argument also shows that asymptotic extremality is concentrated in 2-groups and yields the same sharp exponential rate for the minimum possible index of an abelian subgroup. In graph-theoretic terms, the result determines the sharp asymptotic chromatic-versus-clique growth rate for noncommuting graphs of groups. This result resolves Erdős Problem #117 at the level of its sharp exponential asymptotics.