Coset correct means on groups and the probability that two elements commute
Pith reviewed 2026-05-17 04:14 UTC · model grok-4.3
The pith
The probability that two random group elements commute is positive exactly for finite-by-abelian-by-finite groups when measured with coset-correct means.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Coset correct means are finitely additive means on groups that assign equal weight to the cosets of every subgroup. Every group has a coset correct mean, constructed using B. H. Neumann's theorem via ultrafilters or Hahn-Banach. The degree of commutativity is defined as the mean of the set of pairs that commute and is proven independent of the choice of such mean. This degree is positive precisely when the group is finite-by-abelian-by-finite. The defect of a mean is defined and shown to be either 0 or 1 for the group, with 0 exactly when the group is amenable.
What carries the argument
Coset correct means (CCMs), finitely additive means assigning equal weight to all cosets of any subgroup, used to define a choice-independent degree of commutativity.
If this is right
- The degree of commutativity provides a well-defined probabilistic invariant for all groups.
- It unifies and recovers previous characterizations of groups with positive commuting probability.
- The defect of a group is 0 or 1, with 0 if and only if the group is amenable.
- CCMs allow the study of commutativity probabilities even in non-amenable groups.
Where Pith is reading between the lines
- Other group properties might be definable using averages over CCMs in a choice-independent way.
- This classification might help in understanding the structure of groups with bounded commuting probability.
- Explicit calculations of the degree could be done for concrete examples like symmetric groups or free products.
- One could test the result by verifying the degree for groups known to be finite-by-abelian-by-finite.
Load-bearing premise
The constructions of coset correct means for arbitrary groups via the Ultrafilter Lemma and Hahn-Banach Theorem, relying on B. H. Neumann's theorem, are valid and yield a well-behaved, choice-independent degree of commutativity.
What would settle it
Finding a group that is not finite-by-abelian-by-finite yet has a positive degree of commutativity for some coset correct mean, or a finite-by-abelian-by-finite group with zero degree, would disprove the claim.
read the original abstract
Amenable groups are those admitting an invariant mean -- a finitely additive probability mean that assigns equal ``weight'' to any two translates of the same set. We introduce coset correct means (CCMs), a class of finitely additive means that, for any subgroup, assigns equal weight to all its cosets, weakening and therefore generalising the notion of an invariant mean. We show that, unlike the case for invariant means, every group admits a CCM and give two constructions -- one via the Ultrafilter Lemma and one via the Hahn--Banach Theorem -- both relying on a Theorem of B. H. Neumann. Using CCMs, we define a degree of commutativity for arbitrary groups, measuring the ``probability'' that two random elements of a group commute. We prove that this degree of commutativity is independent of the choice of CCM and is positive precisely for finite-by-abelian-by-finite groups, recovering and unifying previous characterisations. We also introduce a defect function that quantifies the failure of left invariance for finitely additive means, and define the defect of a group as the infimum of these. We then prove a dichotomy: the defect for a group is either 0 or 1, with 0 characterising amenable groups.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces coset correct means (CCMs), finitely additive probability means on groups that assign equal weight to all cosets of any subgroup, generalizing invariant means. It proves every group admits a CCM via two constructions (Ultrafilter Lemma and Hahn-Banach theorem, both using B. H. Neumann's theorem). It defines a degree of commutativity as the CCM-measure of the set of commuting pairs in G×G, proves this degree is independent of the chosen CCM and positive precisely when G is finite-by-abelian-by-finite (recovering prior characterizations), and introduces a defect function whose infimum is either 0 or 1, with 0 characterizing amenable groups.
Significance. If the results hold, the work supplies a canonical, choice-independent invariant measuring the probability that two elements commute in arbitrary groups, unifying earlier results that were limited to amenable or other special cases. The explicit constructions of CCMs for all groups and the defect dichotomy are useful additions to the study of finitely additive means. The independence claim, if rigorously established for arbitrary CCMs, would be a notable strength allowing a well-defined notion without reference to a specific mean.
major comments (2)
- [Independence of the degree of commutativity] § on independence of the degree of commutativity: The central claim requires that for any CCM m the value m({(g,h) | gh = hg}) is the same. The constructions establish existence of some CCMs but do not automatically force agreement on the commuting set for every possible CCM. An explicit argument is needed showing that any two CCMs must assign identical measure to this set (e.g., by proving the commuting set is a union of cosets or has some other property forcing equality under the coset-correct condition). Without this, the independence and the subsequent characterization may fail to hold for all CCMs.
- [Characterization of groups with positive commutativity degree] Theorem characterizing groups with positive degree: The proof that the degree is positive precisely for finite-by-abelian-by-finite groups must verify that the CCM-based definition recovers the known characterizations without circularity or dependence on the specific constructions. If the argument uses properties true only for the constructed CCMs rather than all CCMs, the unification claim would need adjustment.
minor comments (2)
- [Abstract] The abstract could briefly name one or two prior characterizations being recovered to give readers immediate context.
- [Definition of the defect function] Notation for the defect function and its infimum should be introduced with a clear displayed definition to avoid ambiguity in later sections.
Simulated Author's Rebuttal
We thank the referee for the thorough reading and valuable feedback on our manuscript. The comments highlight important points regarding the rigor of our independence claim and characterization. We address each major comment below and will revise the manuscript to strengthen the presentation where needed.
read point-by-point responses
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Referee: [Independence of the degree of commutativity] § on independence of the degree of commutativity: The central claim requires that for any CCM m the value m({(g,h) | gh = hg}) is the same. The constructions establish existence of some CCMs but do not automatically force agreement on the commuting set for every possible CCM. An explicit argument is needed showing that any two CCMs must assign identical measure to this set (e.g., by proving the commuting set is a union of cosets or has some other property forcing equality under the coset-correct condition). Without this, the independence and the subsequent characterization may fail to hold for all CCMs.
Authors: We agree that an explicit, self-contained argument for independence across all CCMs strengthens the paper. In the current manuscript, independence is derived from the coset-correct property applied to the centralizer subgroups in G×G, which ensures that the measure of the commuting set is fixed once the mean respects cosets of all subgroups. However, we acknowledge that this could be stated more directly without relying on the specific constructions. We will add a dedicated lemma in the revision proving that the commuting set is a union of cosets of a suitable normal subgroup in G×G, forcing any CCM to assign it the same value. This addresses the concern directly. revision: yes
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Referee: [Characterization of groups with positive commutativity degree] Theorem characterizing groups with positive degree: The proof that the degree is positive precisely for finite-by-abelian-by-finite groups must verify that the CCM-based definition recovers the known characterizations without circularity or dependence on the specific constructions. If the argument uses properties true only for the constructed CCMs rather than all CCMs, the unification claim would need adjustment.
Authors: The characterization proof relies only on the general definition of a CCM and the coset-correct property, not on any special features of the ultrafilter or Hahn-Banach constructions. We show that a positive degree implies the existence of a finite-index abelian subgroup by using the fact that a CCM must assign positive measure to cosets in a way that forces the centralizer to have finite index, recovering the known result. To eliminate any potential perception of circularity, we will insert a short paragraph in the revision explicitly noting that all steps hold for an arbitrary CCM and do not invoke the existence constructions beyond the fact that at least one CCM exists. revision: yes
Circularity Check
No circularity: independence and characterization rest on external theorems and explicit proof
full rationale
The paper constructs CCMs via the Ultrafilter Lemma and Hahn-Banach Theorem, both relying on B. H. Neumann's theorem (external results). It then proves that the degree of commutativity, defined as m({(g,h) | gh=hg}) for a CCM m, is independent of the specific CCM chosen and positive exactly for finite-by-abelian-by-finite groups. This independence is asserted as a proved statement rather than a definitional or fitted reduction. No self-citation load-bearing the central claim, no ansatz smuggled in, and no renaming of known results as new derivations. The constructions establish existence while the independence and dichotomy are separate arguments against external benchmarks, making the derivation self-contained.
Axiom & Free-Parameter Ledger
axioms (3)
- standard math Ultrafilter Lemma
- standard math Hahn-Banach Theorem
- domain assumption B. H. Neumann's Theorem
invented entities (3)
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coset correct mean (CCM)
no independent evidence
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degree of commutativity
no independent evidence
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defect function
no independent evidence
discussion (0)
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