Higher Commutativity in Finite Groups: Exact Asymptotics and Finite Spectrum

Robert Shwartz, Vadim E. Levit

Pith reviewed 2026-05-08 18:44 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords abelianfiniteordercommutingexacthighermaximumspectrum

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The pith

The number of homomorphisms from the free abelian group of rank r into a finite group G grows asymptotically as k * m^r, where m is the order of the largest abelian subgroup and k is the number of such subgroups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Finite groups consist of elements that may or may not commute with each other. When picking multiple elements at random, the chance they all commute with one another is a higher commuting probability. This paper focuses on the exact way these probabilities behave when the number of elements r becomes large. It turns out that the behavior is controlled by the biggest abelian subgroup inside the group, meaning a subgroup where every pair of elements commutes. The count of ways to choose such commuting r-tuples is dominated by contributions from these largest abelian parts. Additionally, the paper shows that a generating function tracking these counts for all r is a rational function whose expansion involves only a finite number of terms linked to the sizes of abelian subgroups. This finiteness allows the sequence to satisfy a linear recurrence relation, making it possible to compute terms from previous ones. In special cases like cyclic groups, explicit formulas are derived. Overall, this links combinatorial counting in groups to their internal subgroup geometry.

Core claim

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.

Load-bearing premise

That the asymptotic is dominated exactly by the maximal abelian subgroups without significant contributions from smaller ones or other structures for the leading term, and that the rank-generating series admits a finite spectrum supported precisely on the abelian subgroup indices.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an exact leading asymptotic for commuting r-tuples in finite groups, with growth rate set by the largest abelian subgroup and coefficient equal to their count.

read the letter

The main result is that the number of homomorphisms from the free abelian group of rank r into a finite group G grows as c m^r plus lower-order terms, where m is the size of the largest abelian subgroup and c is the number of them. This follows from the fact that all commuting r-tuples lie in the union of A^r over maximal abelian subgroups A, combined with inclusion-exclusion: intersections of distinct maximal ones sit inside strictly smaller subgroups, so their contributions have smaller exponential base and vanish in the leading term. The r-th root of the higher commuting probability therefore tends to m over |G|.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to prove an exact dominant asymptotic for the number of homomorphisms Hom(ℤ^r, G) from the free abelian group of rank r to a finite group G. Specifically, this number is c m^r + O(λ^r) where m is the maximum order of an abelian subgroup of G, c is the number of abelian subgroups of order m, and λ < m. As a consequence, the r-th root of the higher commuting probability (the probability that r random elements commute pairwise) tends to m/|G|, and the r-th root of the number of simultaneous conjugacy classes of such commuting r-tuples tends to m. The paper further asserts that the rank-generating series associated to these counts is rational and admits a finite Dirichlet-spectrum expansion supported on the indices of abelian subgroups of G. This implies a finite linear recurrence for the sequence, a finite-rank Hankel matrix, and an inverse finite-spectrum theorem stating that the tail of the hierarchy determines the full spectrum. Additional results include expressions for the dominant base in split abelian extensions via fixed-subgroup geometry and exact subgroup-lattice formulas for abelian acting quotients, with closed formulas in the cyclic and coprime cases.

Significance. If these results hold, they offer a significant advancement in the study of higher commutativity in finite groups by providing exact leading asymptotics tied directly to the maximal abelian subgroups rather than approximate bounds. The finite spectrum property is a strong feature, as it furnishes a rational generating function and recurrence relations that facilitate exact computations and asymptotic analysis for all r. The inverse theorem, allowing recovery of the spectrum from the tail, is particularly elegant and could have broader implications for poset-based counting in group theory. The derivations appear to rely on inclusion-exclusion over the finite poset of abelian subgroups, which is a clean and standard technique in the field.

minor comments (2)

[Introduction] In the introduction, the precise definition of the rank-generating series and its relation to the Dirichlet spectrum could be stated explicitly with the relevant generating function formula to aid readability.
[The spectrum theorem] The statement of the inverse finite-spectrum theorem would benefit from a brief remark on how the finite number of distinct abelian subgroup orders guarantees the rationality of the generating function.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main results, and recommendation to accept the manuscript. We are pleased that the exact asymptotics for Hom(ℤ^r, G), the finite Dirichlet spectrum, the inverse theorem, and the applications to split extensions and abelian quotients were viewed as significant advancements.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard poset inclusion-exclusion

full rationale

The central asymptotic for the number of homomorphisms from the free abelian group of rank r follows directly from the finite poset of abelian subgroups of G. Inclusion-exclusion on the union over maximal-order abelian subgroups A yields a leading term exactly equal to (number of such A) times m^r, where m is the maximum order, because all proper intersections have strictly smaller order and thus contribute only to lower-order terms. The finite Dirichlet spectrum of the rank-generating series is an immediate consequence of there being only finitely many distinct abelian subgroup orders in a finite group, producing a rational function and linear recurrence without any self-definitional, fitted-prediction, or self-citation load-bearing steps. All claims rest on external group-theoretic invariants (subgroup orders and lattice structure) rather than reducing to the paper's own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims depend on the finiteness of G and standard homomorphism properties, with the finite spectrum being a key derived or assumed property in the abstract.

axioms (2)

domain assumption G is a finite group.
The entire study is restricted to finite groups as per the title and abstract.
standard math Homomorphisms from free abelian groups correspond to commuting tuples.
This is a standard fact in group theory used to count commuting elements.

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