REVIEW 4 minor 6 references
The Ingleton inequality holds for metacyclic groups and fails for supersoluble groups
T0 review · 0 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read This paper proves that metacyclic groups never violate the Ingleton inequality, and that infinitely many supersoluble groups do.
desk verdict A clean, checkable resolution of the metacyclic case plus the first supersoluble counterexamples; the census is a bonus, not the core. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the Ingleton offender itself: a quadruple $(H_1,H_2,H_3,H_4)$ of subgroups of a finite group $G$ for which the order inequality $|H_1||H_2||H_{34}||H_{123}||H_{124}| \ge |H_{12}||H_{13}||H_{14}||H_{23}||H_{24}|$ fails. The paper's central group-theoretic tools are: the product criterion of Proposition 2.6, which shows that if $H_{12} = H_{123}H_{124}$ the inequality always holds and hence in any offender $H_1$ and $H_2$ cannot be cyclic; and the normal-subgroup replacement argument of Theorem 1.3, which replaces $H_i$ by $NH_i$ and tracks how each side of the inequality changes, showing offenders survive unless $N$ meets all $H_i$ trivially. The supersoluble counterexamples are constructed explicitly as matrices in the Borel subgroup of $\mathrm{SL}_3(p)$, with the four subgroups chosen so that three pairwise intersections have order $p-1$, one has order $2$, and $H_{34}=1$, making the right-hand side exceed the left for $p\ge 5$.
What would settle it
Run the supplied Magma code on the explicit five-matrix construction for $p=5$: if the product $p^2(p-1)^2$ fails to be at least $2(p-1)^4$, the supersoluble violator is real; conversely, a computer search over all metacyclic groups of order below 1024 that turns up any Ingleton offender would refute Theorem 1.1.
Extended reading notes
Core claim
The central claim is a dichotomy among soluble groups: metacyclic groups are Ingleton-clean, while supersoluble groups are not. Concretely, Theorem 1.1 asserts that no quadruple of subgroups of a metacyclic group can make the Ingleton inequality fail, and Theorem 1.2 exhibits, for every odd prime $p\ge 5$, a supersoluble group of order $p^3(p-1)$ (the semidirect product of an extraspecial group of order $p^3$ and exponent $p$ by a cyclic group of order $p-1$) containing a generative Ingleton offender. The normal-subgroup dichotomy (Theorem 1.3) states that in any offender, a normal subgroup either is disjoint from all four subgroups or quotients to a new offender, and this is the key to the metacyclic proof. A computational classification (Theorem 1.4) enumerates all Ingleton violators of order less than 1024.
Load-bearing premise
The completeness of the exhaustive classification for groups of order below 1024 depends on the Magma SmallGroups database and its subgroup enumeration being correct and exhaustive; the hand-written theorems about metacyclic and supersoluble groups do not depend on this assumption.
Editorial extensions
If this is right
- The metacyclic case is closed: the earlier claimed proof had a gap, and Theorem 1.1 provides a correct argument that all metacyclic groups satisfy the Ingleton inequality.
- Supersoluble groups, which sit immediately above nilpotent groups in the soluble hierarchy, contain infinitely many generative Ingleton violators for every odd prime $p\ge 5$.
- Any minimal nilpotent Ingleton violator, if it exists, must consist only of indomitable offenders; the exhaustive search confirms no nilpotent violator exists below order 1024.
- All Ingleton violators of order at most 1023 are classified: 391 violators total, of which 61 are irreducible, so violations are rare among small groups.
Reading between the lines
- Because the matrix construction works for prime powers $q$ and the $q=4$ case already produces the non-supersoluble order-192 violator, the family of examples may extend farther up the solubility hierarchy than the theorems state; checking larger prime powers is a natural test.
- The complete absence of nilpotent and odd-order violators below order 2000 suggests that the prime 2 and non-nilpotent structure are essential to any Ingleton violation, but the paper leaves the nilpotent and odd-order questions open.
- Theorem 1.3 isolates the genuinely hard case for a nilpotent proof: a normal subgroup that is disjoint from all four subgroups of an offender. The supersoluble examples show this case can occur, so any nilpotent proof must rule it out by purely nilpotent structure.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies Ingleton's inequality for finite groups, which asserts a certain product inequality for the orders of intersections of four subgroups. The main results are that no metacyclic finite group contains an Ingleton offender (Theorem 1.1), that for every prime p at least 5 the supersoluble group p^{1+2}_+ ⋊ C_{p-1} contains a generative Ingleton offender and hence violates the inequality (Theorem 1.2), and a general normal-subgroup reduction theorem (Theorem 1.3). The paper also reports an exhaustive Magma search showing that all Ingleton violators of order less than 1024 are contained in Tables 1 and 2 (Theorem 1.4), and it gives structural restrictions on minimal violators. The proofs of Theorems 1.1 and 1.2 are self-contained; Theorem 1.4 is a computational classification whose completeness rests on the SmallGroups database.
Significance. If correct, the paper settles the metacyclic case completely and sharply locates the boundary of the Ingleton inequality between metacyclic and supersoluble groups, which is a significant step given that the nilpotent case remains open. The explicit matrix construction in Section 4 is parameter-free and checkable by hand, and the normal-subgroup theorem provides a useful reduction tool for future work on nilpotent groups. The supplied Magma package supports reproducibility of the computational census, and the paper's list of open questions is valuable. The main theoretical claims do not depend on the computational part, and Remark 3.1 accurately diagnoses why the same method cannot settle the nilpotent case.
minor comments (4)
- [Abstract and Introduction] The typos 'Inlgeton' and 'metacycle' should be corrected to 'Ingleton' and 'metacyclic'.
- [Section 4, Lemma 4.1] The sentence introducing the last conjugation says 'we conjugate the matrix u3 by the matrix h2', but the matrix displayed in the calculation is u2; the text should say u2, since the subsequent power computation concerns u2.
- [Section 3, proof of Theorem 1.3] The case analysis is terse at the point where N is assumed to lie in H4 but not in H2: in that case the cancelled factor is |H14|, not |H12|, and the argument is closed by the bound |J124|/|H124| ≤ |J14|/|H14|. Spelling out this case division explicitly would improve readability, although the proof as written is correct.
- [Section 5, Theorem 1.4] The completeness of the classification of Ingleton violators of order below 1024 depends on the correctness and completeness of the SmallGroups database and on Magma's subgroup enumeration routines. Since no independent verification of these routines is described, the statement 'if |G|<1024 and G is an Ingleton violator, then G is known' is a reproducibility caveat rather than a fully formal proof; this does not affect Theorems 1.1 and 1.2, which have hand-written proofs.
Circularity Check
No circularity: The theorems are proved by self-contained group-theoretic arguments, and the computer census is an external reproducibility caveat, not a circular step.
full rationale
The paper's central claims are derived from standard group-theoretic facts and explicit constructions, not from fitted parameters or self-referential definitions. Theorem 1.1 follows from Corollary 2.7 and Theorem 1.3 by an induction on |G|: a cyclic H1 would force H12=H123H124, contradicting Proposition 2.6; in a metacyclic group this makes H1∩K nontrivial, and quotienting by the resulting normal subgroup gives a smaller metacyclic group, so induction applies. Theorem 1.2 is verified by explicit matrices: the subgroups Hi are Frobenius groups of order p(p−1), the intersections are computed directly, and the inequality fails precisely when p^2 < 2(p−1)^2, i.e., for every prime p≥5. Theorem 1.3 is proved by tracking how each side of the Ingleton inequality changes when a normal subgroup is absorbed into one of the Hi. No parameter is fitted to data, no theorem is imported from the authors' prior work, and the criticisms of Oggier and Stancu are about a gap in their proof, not a reliance on it. The exhaustive small-order census depends on the correctness of the Magma SmallGroups database and subgroup enumeration, but this is a reproducibility caveat about external computational tools, not circular reasoning. The paper is self-contained against the claims that matter, so the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- domain assumption The Magma SmallGroups library contains a complete and correct list of all groups of order at most 1023, and the subgroup enumeration algorithm is exhaustive.
- standard math Every subgroup of a cyclic group is characteristic (hence normal in any group in which the cyclic group is normal).
- standard math Standard results on subgroup orders and intersections (e.g., the order formula |AB| = |A||B|/|A∩B|) hold as stated.
Cite this review
Pith. "Pith review of The Ingleton inequality holds for metacyclic groups and fails for supersoluble groups." pith.science (2026). https://pith.science/paper/BG7AGDOC
@misc{pith2026250522565,
author = {Pith},
title = {Pith review of: The Ingleton inequality holds for metacyclic groups and fails for supersoluble groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/BG7AGDOC}},
note = {Machine review of arXiv:2505.22565}
}
abstract
The Ingleton inequality first appeared in matroid theory, where Ingleton proved in 1971 that every rank function coming from a representable matroid on four subsets satisfies a particular inequality. Because this inequality is not implied by submodularity, Shannon-type axioms alone, it and various analogues play a central role in separately linear and non-linear phenomena in a variety of areas of mathematics. The Ingleton inequality for finite groups concerns the various intersections of four subgroups. It holds for many quadruples of subgroups of finite groups, but not all, the smallest example being four subgroups of $S_5$, of order 120. Open questions are whether the Inlgeton inequality always holds for metacycle and nilpotent groups. (There is a proof in the literature due to Oggier and Stancu, but there is an already known issue with their proof, which we address in this article.) In this paper we prove that the Ingleton inequality always holds for metacycle groups, but that it fails for supersoluble groups, a class of groups only a little larger than nilpotent groups. Although we do not resolve the nilpotent case here we do make some reductions, and also prove that there are no nilpotent violators of the Ingleton inequality of order less than 1024. We end with a list of Ingleton inequality violating groups of order at most 1023. The article comes with a Magma package that allows reproduction of all results in the paper and for the reader to check the Ingleton inequality for any given finite group.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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