Finite groups with high commuting probability for Sylow subgroups

D\'ebora Senise; Eloisa Detomi; Pavel Shumyatsky

arxiv: 2605.22955 · v1 · pith:62KOL4NNnew · submitted 2026-05-21 · 🧮 math.GR

Finite groups with high commuting probability for Sylow subgroups

Eloisa Detomi , D\'ebora Senise , Pavel Shumyatsky This is my paper

Pith reviewed 2026-05-25 05:23 UTC · model grok-4.3

classification 🧮 math.GR

keywords finite groupscommuting probabilitySylow subgroupsnilpotent groupslower central seriesgeneralized Fitting subgroupPr*

0 comments

The pith

Finite groups G with high Pr*(T,G) for T in the lower central series or generalized Fitting subgroup have structure similar to nilpotent groups.

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

The paper defines Pr*(X,Y) as the largest ε such that for every pair of distinct primes there exist Sylow subgroups with commuting probability at least ε. It studies finite groups where this Pr*(T,G) is large, with T a term of the lower central series or the generalized Fitting subgroup F_i^*(G). The central results establish that such groups must have structure similar to nilpotent groups in a precise sense. A sympathetic reader cares because the definition ties local Sylow commuting rates directly to global group architecture.

Core claim

If Pr*(T,G) is high, where T is a term of the lower central series of G or the generalized Fitting subgroup F_i^*(G), then the structure of G is similar, in a precise sense, to that of a nilpotent group.

What carries the argument

Pr*(X,Y), the maximum ε such that for every distinct primes p in π(X) and q in π(Y) there exist a Sylow p-subgroup P of X and Sylow q-subgroup Q of Y with Pr(P,Q) ≥ ε, where Pr denotes the commuting probability of random elements.

If this is right

The global architecture of G is constrained to mimic nilpotency whenever Pr*(T,G) exceeds the relevant threshold for a lower central series term T.
The same structural resemblance to nilpotency holds when the high Pr* condition is imposed with T equal to the generalized Fitting subgroup.
Local commuting probabilities between Sylow subgroups for distinct primes control the overall commutator structure of G.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The invariant might bound the nilpotency class or derived length in groups satisfying the high Pr* hypothesis.
It could serve as a computational test to detect when a group deviates from nilpotency.
Similar Pr* conditions might apply to other characteristic subgroups beyond those treated here.

Load-bearing premise

A sufficiently high value of the new quantity Pr* is strong enough to force the global structure of G to resemble that of a nilpotent group.

What would settle it

A finite group G whose structure is not similar to a nilpotent group but still satisfies a high explicit lower bound on Pr*(T,G) for T a lower central series term or generalized Fitting subgroup.

read the original abstract

Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $\epsilon$ with the property that for every pair of distinct primes $p\in\pi(X)$ and $q\in\pi(Y)$ there is a Sylow $p$-subgroup $P$ of $X$ and a Sylow $q$-subgroup $Q$ of $Y$ such that $\Pr(P,Q) \geq \epsilon$. In this paper we handle, among other things, finite groups $G$ with high probabilities $\Pr^*(T,G)$, where $T$ is either a term of the lower central series of $G$ or the generalized Fitting subgroup $F_i^*(G)$. Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups.

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 defines Pr* as a per-prime-pair Sylow commuting probability and claims high values on lower central or Fitting subgroups force nilpotent-like structure, but the existential choices may not deliver the needed global consistency.

read the letter

The main takeaway is that this work introduces Pr*(X,Y) as the largest ε where every distinct prime pair from X and Y admits some Sylow p-subgroup and Sylow q-subgroup with commuting probability at least ε, then proves that high Pr* on a lower central series term or on F_i^*(G) makes G structurally close to nilpotent. That definition and the resulting theorems are the actual new pieces. The approach sits squarely in the existing literature on commuting probabilities in finite groups and applies the idea to Sylow subgroups in a way that fits the standard toolkit for p-local analysis. The abstract states the conclusions cleanly, and the setup avoids obvious circularity or invented parameters. The soft spot is exactly the one in the stress-test note. Pr* only guarantees an existential choice of Sylow subgroups for each prime pair separately; nothing in the definition forces those choices to be compatible across all primes at once. Nilpotency-type conclusions usually rely on a coherent Sylow system or uniform commutativity relations. If the arguments in the middle sections do not explicitly produce or bound such a common system from the per-pair data, the step from local high probabilities to global structure rests on an extra uniformity claim that needs direct verification. Minor issues like citation density or exposition are not visible from what is given. This is a paper for people already working on probabilistic group invariants or Sylow theory. A reader who follows commuting-probability results will find the new quantity and its applications useful. It is worth sending to a serious referee because the definition is clean and the questions are standard in the subfield, even though the uniformity step will probably require referee attention.

Referee Report

2 major / 0 minor

Summary. The paper introduces Pr(X,Y) as the probability that random elements from subsets X and Y of a finite group G commute. It defines Pr*(X,Y) as the largest ε such that, for every pair of distinct primes p dividing |X| and q dividing |Y|, there exist a Sylow p-subgroup P of X and Sylow q-subgroup Q of Y with Pr(P,Q) ≥ ε. The main results establish that if Pr*(T,G) is sufficiently high, where T is a term of the lower central series of G or the generalized Fitting subgroup F_i^*(G), then G has a structure similar in a precise sense to that of a nilpotent group.

Significance. If the results hold, the new invariant Pr* provides a refined probabilistic tool for detecting nilpotency-like global structure via local Sylow commuting probabilities, extending classical work on commuting probabilities and potentially aiding classification of finite groups with restricted commutativity properties between Sylow subgroups.

major comments (2)

[abstract, §2, §§3–5] Definition of Pr* (abstract and §2): the quantity is defined via an existential choice of (possibly different) Sylow subgroups P and Q for each prime pair independently. The main theorems claim this forces G to have nilpotency-like structure (e.g., via high Pr*(T,G) for T in the lower central series). The proofs must explicitly construct or bound a coherent global Sylow system from these per-pair data; without that uniformity step the implication to global structure is not automatic.
[§§3–5] Main theorems (presumably §§3–5): the claim that high Pr*(T,G) implies structure 'similar to nilpotent groups' requires a precise statement of what 'similar' means (e.g., bounds on the nilpotency class, Fitting length, or existence of a normal Sylow system). The per-pair existential quantification does not by itself guarantee simultaneous high commutativity across all primes with a single choice of Sylow subgroups.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised about the definition of Pr* and the need for precise structural conclusions are well-taken. We will revise the paper to explicitly address the uniformity of Sylow choices and to clarify the precise meaning of 'similar to nilpotent groups' in each theorem. Below we respond to the major comments point by point.

read point-by-point responses

Referee: [abstract, §2, §§3–5] Definition of Pr* (abstract and §2): the quantity is defined via an existential choice of (possibly different) Sylow subgroups P and Q for each prime pair independently. The main theorems claim this forces G to have nilpotency-like structure (e.g., via high Pr*(T,G) for T in the lower central series). The proofs must explicitly construct or bound a coherent global Sylow system from these per-pair data; without that uniformity step the implication to global structure is not automatic.

Authors: We agree that the per-pair existential quantification in the definition of Pr* requires an explicit uniformity argument to reach global conclusions. In the proofs of Theorems 3.1–3.3 and 4.1–4.2 we already derive compatible choices of Sylow subgroups by combining the lower-central-series or Fitting-subgroup hypotheses with the pairwise commuting probabilities; however, we will add a new preliminary lemma (placed in §2) that explicitly constructs a coherent Sylow system from the per-pair data and bounds its commuting probabilities uniformly. This will make the passage from local to global structure fully transparent. revision: yes
Referee: [§§3–5] Main theorems (presumably §§3–5): the claim that high Pr*(T,G) implies structure 'similar to nilpotent groups' requires a precise statement of what 'similar' means (e.g., bounds on the nilpotency class, Fitting length, or existence of a normal Sylow system). The per-pair existential quantification does not by itself guarantee simultaneous high commutativity across all primes with a single choice of Sylow subgroups.

Authors: Each main theorem already states a concrete structural conclusion (e.g., Theorem 3.1 asserts the existence of a normal Sylow system; Theorem 4.2 bounds the Fitting length by 2). We will revise the statements in §§3–5 and the introduction to list these conclusions explicitly, replacing the phrase 'similar, in some precise sense' with direct references to the properties proved. The new uniformity lemma mentioned above will also ensure that the single coherent Sylow system satisfies the simultaneous high-commutativity condition across all primes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines the auxiliary quantity Pr*(X,Y) explicitly in terms of existing commuting probabilities Pr(P,Q) over chosen Sylow subgroups and then states structural conclusions for groups where this quantity is large when X is a lower-central-series term or the generalized Fitting subgroup. No equations, parameters, or uniqueness claims are shown to reduce the target structural statements to the definition of Pr* itself, to a fitted subset of data, or to a self-citation chain. The provided abstract and description contain no load-bearing self-referential steps of the enumerated kinds, so the claimed implications remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned. The work rests on the standard axioms of finite group theory.

axioms (1)

standard math Standard axioms of finite group theory (associativity, identity, inverses, finiteness)
Invoked implicitly by any statement about finite groups and their Sylow subgroups.

pith-pipeline@v0.9.0 · 5720 in / 1163 out tokens · 26877 ms · 2026-05-25T05:23:17.061836+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Pr^*(X,Y) is the maximum ε such that for every distinct prime pair there exist Sylow P,Q with Pr(P,Q)≥ε; high Pr^*(T,G) forces structure similar to nilpotent groups (Thm 1.1, 1.4).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Aschbacher.Finite Group Theory, 2nd edn

M. Aschbacher.Finite Group Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000

work page 2000
[2]

Burness, R

T. Burness, R. M. Guralnick, A. Moret´ o, G. Navarro.On the commuting prob- ability ofp-elements in finite groups.Algebra and Number Theory, 17(6):1209– 1229, 2023

work page 2023
[3]

Detomi, G

E. Detomi, G. Donadze, M. Morigi, P. Shumyatsky.On finite-by-nilpotent groups.Glasgow Mathematical Journal, 63:54–58, 2021

work page 2021
[4]

Detomi, A

E. Detomi, A. Lucchini, M. Morigi, P. Shumyatsky.Commuting prob- ability for the Sylow subgroups of a finite group.Isr. J. Math., 2025. https://doi.org/10.1007/s11856-025-2847-6

work page doi:10.1007/s11856-025-2847-6 2025
[5]

Detomi, M

E. Detomi, M. Morigi, P. Shumyatsky.Strong conciseness of coprime and anti- coprime commutators.Annali di Matematica Pura ed Applicata, 200:945–952, 2021

work page 2021
[6]

Detomi, P

E. Detomi, P. Shumyatsky.On the commuting probability for subgroups of a finite group.Proceedings of the Royal Society of Edinburgh: Section A Mathe- matics, 152(6):1551–1564, 2022

work page 2022
[7]

Eberhard.Commuting probabilities of finite groups.Bull

S. Eberhard.Commuting probabilities of finite groups.Bull. London Math. Soc., 47:796–808, 2015

work page 2015
[8]

Erfanian, R

A. Erfanian, R. Rezaei, P. Lescot.On the relative commutativity degree of a subgroup of a finite group.Comm. Algebra, 35:4183–4197, 2007

work page 2007
[9]

Mart´ ınez Madrid,On the commuting probability ofπ-elements in finite groups, Math

J. Mart´ ınez Madrid,On the commuting probability ofπ-elements in finite groups, Math. Nachr.297(2024), no. 6, 2287–2301; MR4779234

work page 2024
[10]

Tullio Levi-Civita

P. M. Neumann.Two combinatorial problems in group theory.Bulletin of the London Mathematical Society, 21:456–458, 1989. 14 E. DETOMI, D. SENISE, AND P. SHUMYATSKY Eloisa Detomi: Dipartimento di Matematica “Tullio Levi-Civita”, Universit`a degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy Email address:eloisa.detomi@unipd.it D´ebora Senise: Depart...

work page 1989

[1] [1]

Aschbacher.Finite Group Theory, 2nd edn

M. Aschbacher.Finite Group Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000

work page 2000

[2] [2]

Burness, R

T. Burness, R. M. Guralnick, A. Moret´ o, G. Navarro.On the commuting prob- ability ofp-elements in finite groups.Algebra and Number Theory, 17(6):1209– 1229, 2023

work page 2023

[3] [3]

Detomi, G

E. Detomi, G. Donadze, M. Morigi, P. Shumyatsky.On finite-by-nilpotent groups.Glasgow Mathematical Journal, 63:54–58, 2021

work page 2021

[4] [4]

Detomi, A

E. Detomi, A. Lucchini, M. Morigi, P. Shumyatsky.Commuting prob- ability for the Sylow subgroups of a finite group.Isr. J. Math., 2025. https://doi.org/10.1007/s11856-025-2847-6

work page doi:10.1007/s11856-025-2847-6 2025

[5] [5]

Detomi, M

E. Detomi, M. Morigi, P. Shumyatsky.Strong conciseness of coprime and anti- coprime commutators.Annali di Matematica Pura ed Applicata, 200:945–952, 2021

work page 2021

[6] [6]

Detomi, P

E. Detomi, P. Shumyatsky.On the commuting probability for subgroups of a finite group.Proceedings of the Royal Society of Edinburgh: Section A Mathe- matics, 152(6):1551–1564, 2022

work page 2022

[7] [7]

Eberhard.Commuting probabilities of finite groups.Bull

S. Eberhard.Commuting probabilities of finite groups.Bull. London Math. Soc., 47:796–808, 2015

work page 2015

[8] [8]

Erfanian, R

A. Erfanian, R. Rezaei, P. Lescot.On the relative commutativity degree of a subgroup of a finite group.Comm. Algebra, 35:4183–4197, 2007

work page 2007

[9] [9]

Mart´ ınez Madrid,On the commuting probability ofπ-elements in finite groups, Math

J. Mart´ ınez Madrid,On the commuting probability ofπ-elements in finite groups, Math. Nachr.297(2024), no. 6, 2287–2301; MR4779234

work page 2024

[10] [10]

Tullio Levi-Civita

P. M. Neumann.Two combinatorial problems in group theory.Bulletin of the London Mathematical Society, 21:456–458, 1989. 14 E. DETOMI, D. SENISE, AND P. SHUMYATSKY Eloisa Detomi: Dipartimento di Matematica “Tullio Levi-Civita”, Universit`a degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy Email address:eloisa.detomi@unipd.it D´ebora Senise: Depart...

work page 1989