Higher Commutativity in Finite Groups: Exact Asymptotics and Finite Spectrum

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.