Higher Commutativity in Finite Groups: Exact Asymptotics and Finite Spectrum

TL;DR

This paper investigates the probabilities of pairwise commuting elements in finite groups, providing exact asymptotics, spectral analysis, and formulas for special cases, revealing deep structural insights.

Contribution

It introduces precise asymptotic formulas for higher commuting probabilities and conjugacy class counts, along with spectral and lattice formulas for finite abelian group extensions.

Findings

Asymptotic probability tends to maximum abelian subgroup order divided by group order.

The rank-generating series is rational with finite Dirichlet-spectrum expansion.

Closed formulas are derived for cyclic and coprime cases.

Abstract

For a finite group G, we study the higher commuting probabilities, namely the probabilities that r randomly chosen elements of G commute pairwise, together with the corresponding numbers of simultaneous conjugacy classes of commuting r-tuples. We prove an exact dominant asymptotic for the number of homomorphisms from the free abelian group of rank r to G. The exponential base is the maximum order of an abelian subgroup of G, and the leading coefficient is the number of abelian subgroups of that order. As a consequence, the r-th root of the higher commuting probability tends to this maximum abelian-subgroup order divided by the order of G, while the r-th root of the orbit count tends to the maximum abelian-subgroup order itself. We also prove that the associated rank-generating series is rational and has a finite Dirichlet-spectrum expansion supported on abelian subgroup indices. This spectrum yields a finite linear recurrence, a finite-rank Hankel matrix, and an inverse finite-spectrum theorem: the tail of the hierarchy determines the full abelian-index spectrum. For split abelian extensions, we express the dominant base through fixed-subgroup geometry, and for abelian acting quotients, we obtain an exact subgroup-lattice formula. In the cyclic and coprime cases, this gives closed formulas for all spectral coefficients.