The solution to Kourovka problem 21.88

Basile Beyer de Ryke

The solution to Kourovka problem 21.88

T0 review · 0 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper proves that no finite group of odd order has commuting probability $1/17$, and reduces the next unresolved case $p=97$ to a rigid semidirect-product form.

desk verdict Solid solution to a Kourovka problem: the structural theorem is real, the p=17 exclusion checks out, and the p=97 reduction is honest, so this deserves a serious referee. read the letter →

arxiv 2608.03003 v1 pith:4IDNYX4K submitted 2026-08-04 math.GR

Basile Beyer de Ryke This is my paper · ORCID

classification math.GR MSC 20D6020C1520D10

keywords commutingprobabilityconjugacyclassesgroupsofoddordercoprimeactionSylowp-subgroupKourovkaNotebookProblem21.88Burnsidecongruence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper answers an open problem from the Kourovka Notebook: no finite group of odd order can have commuting probability $1/17$, where the commuting probability is the fraction of ordered pairs of elements that commute. The supporting structural theorem is broader: if an odd prime $p$ occurs as the commuting probability of a finite odd-order group, then a Sylow $p$-subgroup of the group is normal and abelian. A classical congruence then excludes $cp(G)=1/p$ for every odd prime $p<97$, leaving $p=97$ as the next case. The paper does not settle $p=97$, but it compresses any hypothetical example into a narrow structural form with an exact numerical equation.

What carries the argument

The load-bearing identity is $cp(G)=k(G)/|G|$, which turns commuting probability into a count of conjugacy classes. The proof is carried by character-counting inequalities: a strict lower bound $[X:Y]^{-2}cp(Y)<cp(X)\le cp(Y)$ for subgroups, a product bound for normal subgroups, and a solvable-group bound comparing $cp(X)$ with the number of orbits of a Hall $\pi'$-subgroup on $O_{\pi'}(X)$. The second main tool is the Clifford-theoretic orbit formula $k(G)=\sum_{\lambda\in R} k(H_\lambda)$, which counts the irreducible characters of the semidirect product by summing the class numbers of stabilizers over an orbit of linear characters of the abelian Sylow subgroup. These are paired with fixed-point estimates $[P:C_P(B)]\ge p^d$ for subgroups $B$ of prime order acting on the abelian Sylow subgroup.

What would settle it

A direct disproof would be a single finite group of odd order in which exactly $1/17$ of the ordered pairs commute. A weaker but still decisive observation would be an odd-order group with $cp(G)=1/p$ for an odd prime $p$ whose Sylow $p$-subgroup is not normal and abelian; a computer search over small odd-order groups looking for either configuration would test the main claim.

Watch

Extended reading notes

Core claim

The central claim is that the value $1/17$ cannot occur for odd-order groups. More generally, Theorem 1.1 asserts that whenever $p$ is an odd prime and a finite group $G$ of odd order satisfies $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian. Combining this with the congruence $|G|-k(G)\equiv 0\pmod{16}$ for odd-order groups, the paper excludes $1/p$ for every odd prime $p<97$. For $p=97$, the reduction theorem says a least-order example would have the form $G=P\rtimes H$ with $P$ abelian, $A=H/C_H(P)\cong C_3$, $K=C_H(P)$ nonabelian, $C_P(A)=1$, and the residual identity $k/m+8f/(97em)=9/97$ relating the class numbers and character fixed points.

Load-bearing premise

The argument rests on the quoted inequalities for commuting probabilities, especially the strict subgroup bound and the solvable-group orbit bound, being valid in full generality for finite groups of odd order; if one of those cited results carries a hidden restriction, the structural theorem and the exclusion of $1/17$ would collapse.

Editorial extensions

If this is right

No finite group of odd order has commuting probability $1/17$, so the stated Kourovka Notebook problem is closed in the negative.
Any odd-order group with commuting probability $1/p$ for an odd prime $p$ must have a normal abelian Sylow $p$-subgroup, so such groups are semidirect products $P\rtimes H$ with a faithful action of $H$ on $P$.
For every odd prime $p<97$, $1/p$ is not the commuting probability of a finite odd-order group; only the case $p=17$ survives the congruence, and it is excluded directly.
A hypothetical least-order group with probability $1/97$ would satisfy $A\cong C_3$, a nonabelian kernel $K$, $C_P(A)=1$, and the exact residual equation $k/m+8f/(97em)=9/97$; moreover $3$ divides $|K|$ and $K$ is not extraspecial.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same structural theorem suggests a broader restriction: for odd-order groups, primes $p$ with $cp(G)=1/p$ must satisfy $p\equiv1\pmod{16}$, and the only candidate below $97$ was $p=17$, already excluded; this is a natural extension, not a claim proved in the paper.
The $p=97$ reduction funnels the problem into a separate open question: whether any odd-order group has commuting probability $3/35$. If no such group exists, the paper's framework would exclude $1/97$ as well.
The fixed-point and orbit estimates could be turned into a finite search: enumerate coprime semidirect products $P\rtimes A$ with $P$ abelian of order $97^e$ and small acting group $A$, and check whether the exact orbit sum can reach $1/97$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Basile: this one is worth your time. The paper answers Kourovka 21.88 with a clean negative: no finite group of odd order has commuting probability 1/17. The engine is Theorem 1.1, which says that for any odd prime p, cp(G)=1/p forces a normal abelian Sylow p-subgroup. That theorem is new, interesting, and the proof is careful. The character-orbit inequality in Proposition 4.3 is a genuinely useful tool, and the case analysis for p=17 is tight: the pair-counting bound gives a^2≤18, and the two alternatives (abelian K forcing a^2>17, nonabelian K forcing a^2<9) leave no room. I checked the key steps against the cited Guralnick–Robinson results, including the interpretation of (2.6) and the use of (2.1), and I found no misapplication.

What I like most is the restraint. The p=97 case is not settled, and the paper says so plainly; it gives a structural reduction with genuinely restrictive constraints (A=C3, K nonabelian, the residual equation (5.10)) but does not pretend to close it. That is how you report partial progress on an open problem. The Burnside congruence corollary for p<97 is a nice byproduct.

Soft spots? Only minor. The paper leans on the Feit–Thompson theorem at several points, which is heavy but unavoidable given the state of the art. The proofs of (2.3)–(2.5) are sketched rather than fully derived, but they are standard and easily filled. The p=97 section is dense and will take a reader some effort, but it is coherent. I did not see any circularity, parameter fitting, or overclaiming. The citation pattern is appropriate, and the open cases are explicitly flagged.

Who is this for? Group theorists working on commuting probabilities or on the Kourovka Notebook, and anyone who wants to see a masterclass in how to structure a hard classification argument without overselling it. The main theorems are formally stated, the arithmetic is reproducible, and the reduction at 97 is an honest invitation to further work.

Recommendation: send it to a serious referee. Even if the p=97 part remains open, the answer to 21.88 and the structural theorem justify publication in a good group theory journal. I would engage with it and cite it.

Referee Report

0 major / 3 minor

Summary. The paper proves that no finite group of odd order has commuting probability 1/17, thereby giving a negative answer to Kourovka Notebook Problem 21.88. The proof has three main parts. First, a structural theorem (Theorem 1.1) shows that if an odd-order group G satisfies cp(G)=1/p for an odd prime p, then a Sylow p-subgroup of G is normal and abelian; the proof uses the Guralnick-Robinson inequalities, the Feit-Thompson theorem, and a class-counting argument for a semidirect product. Second, for p=17, the paper combines the structural theorem with a character-orbit fixed-point inequality (Proposition 4.3) to reach a contradiction in the least-order counterexample, distinguishing the cases where the complement kernel K is abelian or nonabelian. Third, Burnside's congruence for groups of odd order implies that any such prime p must satisfy p≡1 mod 16, which excludes all odd primes p<97 except 17. A final section studies the remaining case p=97 and reduces it to restricted structural conditions, explicitly leaving the existence question open.

Significance. The main result resolves a long-standing problem in the study of commuting probabilities and is a substantial contribution to the area. The structural theorem (Theorem 1.1) is a strong new statement about odd-order groups with cp(G)=1/p, and the character-orbit bound of Proposition 4.3 is a useful technique that yields the p=17 exclusion cleanly. The paper is careful and largely self-contained: the argument relies on standard cited results (Feit-Thompson, Schur-Zassenhaus, Clifford theory, Guralnick-Robinson inequalities) and does not appear to fit parameters or reduce the target claim to itself. The honesty about the open p=97 case is also a strength. If the proof is correct, the result will likely be of interest to group theorists and to researchers in probabilistic group theory. The main theorem is machine-checkable in structure, though the paper itself does not ship code.

minor comments (3)

[§5.1, Eq. (5.1)] The displayed inequality r^2(97^{d−1}−1) ≤ 97^{d−1} is not equivalent to the preceding bound 1/97 ≤ (97^d + r^2 − 1)/(r^2 97^d); the correct right-hand side is 97^d − 1. The subsequent deduction r ≤ 7 uses the corrected inequality, so the statement of Lemma 5.2 is unaffected, but the displayed equation should be fixed.
[§5, Theorem 5.1 and §5.3] The notation '97em' in the displayed equations is ambiguous and should be typeset as 97^e m (with n = 97^e). The same clarification is needed in the residual equations (5.10)–(5.12).
[§3.3, Lemma 3.3] In the exclusion of A=1, the sentence 'If one factor is abelian, the other would have commuting probability 1/p' is terse; for the case where the p'-factor is abelian, one should explicitly note that cp(H)=1/p would force p to divide |H|, which is impossible.

Circularity Check

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No significant circularity: the p=17 exclusion is derived from external standard theorems and contains no fitted-input or self-citation reduction.

full rationale

The paper's derivation chain is self-contained against external mathematical results and does not reduce to its own inputs. Theorem 1.2 follows from Theorem 1.1, which is proved in Section 3 using the Guralnick-Robinson inequalities (2.1)-(2.7), the Feit-Thompson theorem, Schur-Zassenhaus, and standard character theory from Isaacs. These are independent established results, not results of the present paper, and none of them is defined in terms of the target claim. The key estimates in Sections 3 and 4 are derived from explicit inequalities: for instance, Lemma 3.1 counts conjugacy classes in W⋊P directly and obtains cp(W⋊P)<1/p without assuming the theorem; Proposition 3.2 applies the external bound (2.6) to force b≥a−1; and Proposition 3.4 uses character-extension and Clifford theory to derive cp(R)<1/p. The p=17 arithmetic in Section 4.2 is a genuine contradiction from the character-orbit inequality (4.2): the pair-counting gives a^2≤18, while the abelian-K case forces a^2>17 and the nonabelian-K case forces a^2<9, with no odd a>1 surviving. No parameter is fitted to data, no predicted quantity is identical by construction to an input, and no load-bearing step is justified by a self-citation. The paper's Section 5 on p=97 is explicitly labelled as open and does not affect the negative answer to Kourovka Problem 21.88. The most external component, the Guralnick-Robinson inequalities, is standard published work by other authors and is applied correctly, so its use does not constitute circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The results depend on standard external theorems: Feit-Thompson solvability, Schur-Zassenhaus, Guralnick-Robinson inequalities, Burnside's congruence, and character extension theorems. These are cited and used as black boxes.

assumptions (6)

standard math Feit-Thompson theorem: every group of odd order is solvable.
Invoked in Lemma 3.1, Prop 3.2, and Lemma 5.3 to apply solvable-group character theory.
standard math Guralnick-Robinson inequalities (2.1)-(2.7), including the strict subgroup inequality and the solvable bound (2.6).
Cited from [4,5]; used throughout Sections 3 to 5.
standard math Schur-Zassenhaus theorem for existence of Hall complements.
Used to write G=P⋊H and H=K⋊<alpha> in Sections 3 and 5.
standard math Burnside congruence: |X|-k(X) is divisible by 16 for groups of odd order.
Used in Prop 4.5 and Corollary 1.3 to force p congruent to 1 mod 16.
standard math Clifford theory and Gallagher's theorem for character extension.
Used in Prop 4.3 and in equations (5.8) to (5.9) of Section 5.3.
standard math Normal Sylow criterion from Guralnick-Robinson: cp(X)>1/q implies a normal Sylow q-subgroup.
Used in Proposition 5.6.

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Cite this review

Pith. "Pith review of The solution to Kourovka problem 21.88." pith.science (2026). https://pith.science/paper/4IDNYX4K

@misc{pith2026260803003,
  author       = {Pith},
  title        = {Pith review of: The solution to Kourovka problem 21.88},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IDNYX4K}},
  note         = {Machine review of arXiv:2608.03003}
}

abstract

We give a negative answer to Kourovka Notebook Problem 21.88: no finite group of odd order has commuting probability $1/17$. This follows from a structural theorem asserting that, whenever $p$ is an odd prime and $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian. Together with Burnside's congruence for the number of conjugacy classes of a group of odd order, this also excludes $cp(G)=1/p$ for every odd prime $p<97$. We further study the next unresolved case not excluded by this congruence, namely $p=97$.

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Reference graph

Works this paper leans on

10 extracted references · 9 canonical work pages

[1]

S. M. Buckley and D. MacHale,Conjugate deficiency in finite groups, Irish Math. Soc. Bull. 71(2013), 13–19

work page 2013
[2]

Feit and J

W. Feit and J. G. Thompson,Solvability of groups of odd order, Pacific J. Math.13(1963), 775–1029

work page 1963
[3]

P. X. Gallagher,The number of conjugacy classes in a finite group, Math. Z.118(1970), 175–179

work page 1970
[4]

R. M. Guralnick and G. R. Robinson,On the commuting probability in finite groups, J. Algebra300(2006), no. 2, 509–528

work page 2006
[5]

On the commuting probability in finite groups

R. M. Guralnick and G. R. Robinson,Addendum to paper “On the commuting probability in finite groups”, J. Algebra319(2008), no. 4, 1822

work page 2008
[6]

W. H. Gustafson,What is the probability that two group elements commute?, Amer. Math. Monthly80(1973), no. 9, 1031–1034

work page 1973
[7]

I. M. Isaacs,Character Theory of Finite Groups, Dover Publications, New York, 1994; corrected reprint of the 1976 original

work page 1994
[8]

I. M. Isaacs,Finite Group Theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008

work page 2008

Show all 10 references

[9]

E. I. Khukhro and V. D. Mazurov (eds.),Unsolved Problems in Group Theory: The Kourovka Notebook, 21st ed., Sobolev Institute of Mathematics, Novosibirsk, 2026; arXiv:1401.0300v45, 3 July 2026

2026 arXiv
[10]

MacHale,How commutative can a non-commutative group be?, Math

D. MacHale,How commutative can a non-commutative group be?, Math. Gaz.58(1974), no. 405, 199–202. 12 A venue Hamoir, 1180 Uccle, Belgium Email address:basile.beyer@gmail.com

1974

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

S. M. Buckley and D. MacHale,Conjugate deficiency in finite groups, Irish Math. Soc. Bull. 71(2013), 13–19

work page 2013

[2] [2]

Feit and J

W. Feit and J. G. Thompson,Solvability of groups of odd order, Pacific J. Math.13(1963), 775–1029

work page 1963

[3] [3]

P. X. Gallagher,The number of conjugacy classes in a finite group, Math. Z.118(1970), 175–179

work page 1970

[4] [4]

R. M. Guralnick and G. R. Robinson,On the commuting probability in finite groups, J. Algebra300(2006), no. 2, 509–528

work page 2006

[5] [5]

On the commuting probability in finite groups

R. M. Guralnick and G. R. Robinson,Addendum to paper “On the commuting probability in finite groups”, J. Algebra319(2008), no. 4, 1822

work page 2008

[6] [6]

W. H. Gustafson,What is the probability that two group elements commute?, Amer. Math. Monthly80(1973), no. 9, 1031–1034

work page 1973

[7] [7]

I. M. Isaacs,Character Theory of Finite Groups, Dover Publications, New York, 1994; corrected reprint of the 1976 original

work page 1994

[8] [8]

I. M. Isaacs,Finite Group Theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008

work page 2008