Products of three conjugacy classes in the alternating group

Daniele Dona

REVIEW 2 major objections 5 minor 25 references

Products of three conjugacy classes in the alternating group

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Three conjugacy classes of size at least |Alt(n)|^{1−δ} always multiply to the entire alternating group.

desk verdict Genuine advance with a load-bearing gap: the split-class case of Proposition 3.2 is delegated to a citation, and the constructive alternative offered there does not work as written. read the letter →

arxiv 2505.06012 v1 pith:YR67EQGO submitted 2025-05-09 math.GR math.CO

Daniele Dona This is my paper · ORCID

classification math.GRmath.CO MSC 20B3020E45

keywords alternatinggroupconjugacyclassesproductsofKourovkaNotebookproblemnormalsubsetsfinitesimplegroupsclassstringsconstructiveproof

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

The paper proves that, for sufficiently large n and small δ>0, any three conjugacy classes of Alt(n) of size at least |Alt(n)|^{1−δ} satisfy C1C2C3=Alt(n). This answers Problem 20.23 of the Kourovka Notebook, improves the earlier four-class theorem of Garonzi and Maróti, and combines with known Lie-type results to extend the statement to three normal subsets of every nonabelian finite simple group. The proof avoids character theory entirely: it encodes cycle structures as class strings and reduces the equation α1α2=α3 to a short list of explicitly solved cycle products. Because the reductions are injective and constructive, the argument in principle outputs the three permutations c1,c2,c3 for any given target element.

What carries the argument

The central mechanism is a chain of seven injective reductions θ1,…,θ7 acting on string triples: three class strings, whose ■ symbols mark cycle boundaries, together with a label function that records the information needed to undo each step. Each reduction simplifies the cycle structure while preserving the existence of an aligned solution, meaning a solution α1α2=α3 in which triples of cycles sharing positions also share prescribed common values. The final reduction splits the problem into small subproblems, each solved by one of the explicit templates (12.1)–(12.4), and the solutions are concatenated and pulled back through the inverses of the reductions. The quantitative glue is Proposition 2.3, which translates class size into a bound on the number of cycles and is invoked at every stage to guarantee that shortened classes remain large.

What would settle it

Find n and three conjugacy classes C1,C2,C3 of Alt(n) with |Ci| ≥ |Alt(n)|^{1−δ} and a single permutation g outside C1C2C3; such a triple would refute Theorem 1.2 directly. A cheaper partial check is to test Proposition 2.3 numerically on the extremal class of one (1−δ)n-cycle plus δn fixed points, the class used throughout, to see whether its cycle count stays within (δ+o(1))n; failure there would break the reduction chain.

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Extended reading notes

Core claim

The paper establishes Theorem 1.2: there exist constants δ>0 and n0 such that for all n≥n0 and any three conjugacy classes C1,C2,C3 of G=Alt(n) with |Ci| ≥ |G|^{1−δ}, the product set C1C2C3 equals all of G. The engine is the stronger intermediate Theorem 1.4, which says the same hypotheses imply C1C2⊇C3. The proof works in Sym(n), replacing each class by a class string that fixes an ordering of its cycles, then applying seven injective reductions that progressively simplify the cycle structures, and finally solving the reduced problem by concatenating explicit solutions for four basic types of cycle configurations. Inverting the seven reductions yields the required permutations. Theorem 1.3 follows by combining Theorem 1.2 with the known Lie-type case, giving the same statement for three normal subsets of every nonabelian finite simple group.

Load-bearing premise

The whole chain rests on Proposition 2.3, the quantitative link between class size and number of cycles: if that estimate fails to keep every shortened class nearly maximal after seven rounds of shortening, the argument no longer closes.

Editorial extensions

If this is right

Kourovka Problem 20.23 is answered: three large conjugacy classes suffice to cover Alt(n), where previously four classes were needed.
Theorem 1.3 follows for every nonabelian finite simple group, since the Lie-type case was already known and the alternating case is supplied here, with normal subsets in place of conjugacy classes.
The four-class threshold |G|^{1/2+ε} from Garonzi and Maróti is improved to three classes with exponent 1−δ for Alt(n).
Rodgers' result requiring at most six cycles across three classes is superseded for classes above the size threshold, since the new proof needs only large class size, not a small number of cycles.
Because no character theory is used, the proof yields a constructive algorithm: given g and the three classes, it can output c1,c2,c3 with c1c2c3=g.

Reading between the lines

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

One could extract explicit, though likely far from optimal, values of δ and n0 by tracking constants through the seven reductions, turning the in-principle algorithm into a runnable one.
The class-string reduction scheme is a general device and may apply to other product questions in Sym(n), such as deciding when C1C2 contains a prescribed class for classes of intermediate size, or to two-class covering results for special cycle types.
The author leaves open whether a character-theoretic proof of Theorem 1.2 exists; the constructive nature of this proof suggests that such a proof, if found, would need different quantitative estimates from the usual character-sum bounds.
The tightness discussion implies that the true maximal δ is unknown and probably smaller than 1/2; a natural next step is a numerical study for moderate n to see how large δ can be before three-class coverage fails.

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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

The paper answers Kourovka 20.23: for n large and δ small, any three conjugacy classes of Alt(n) of size at least |G|^{1−δ} multiply to all of G. That is a genuine advance over Garonzi–Maróti's four-class theorem and Rodgers's special cycles, and the proof is elementary throughout, no character theory. The seven reductions in §§5–12 are the real work: each step comes with an explicit map and an inverse walk-back, and the final solutions in §12 are explicit products. I read through the reduction chain and it looks internally coherent; I did not certify every displayed inequality by hand.

The soft spot is Proposition 3.2, the only step that reaches classes of size ≥ |G|^{2δ/3} in the proof of Theorem 1.2. In the split-class case (both C_i ≠ (C_i)^g), the proof says the argument is 'essentially contained' in [GM21, Lemma 5.3(i)], which handles m = n−1,n, and 'the technique works out similarly' for m = n−3,n−2. That is a load-bearing delegation. The offered constructive alternative—apply Theorem 1.4—is not circular, since Thm 1.4 is proven independently later, but it is invalid as written: from |C1||C2| ≥ |G|^{1+δ} you cannot infer that |C1|,|C2|,|O_m| each exceed |G|^{1−δ}; one class might be small. So the split case currently has no proof in this preprint. A referee needs to see the character argument written out for the actual m, or a precise reference.

Minor: in the small-n part of Theorem 1.3, the displayed bound |G|^{1−(n2 log n2)^{-1}} > |G|^{−1} does not force S_i = G; the exponent should be much smaller. Fixable.

Citation pattern is fine: the only imported quantitative tool is Proposition 2.3 from [DMP24]/[GM21], and it is used as a lemma, not a restatement of the result.

Bottom line: this deserves a serious referee. The main theorem is almost certainly true, and the combinatorial machinery is worth publishing once Prop 3.2 is fixed. I would not cite it in my own work until the gap is closed, but I would encourage the editor to send it to review and ask for that revision.

Referee Report

2 major / 5 minor

Summary. The paper proves that for a sufficiently small constant δ and all sufficiently large n, every product C1C2C3 of three conjugacy classes of Alt(n) with |Ci| ≥ |Alt(n)|^{1−δ} equals the whole group (Theorem 1.2), and it extends this to three normal subsets of every non-abelian finite simple group (Theorem 1.3). The main combinatorial work establishes Theorem 1.4, which asserts that two large classes cover a third, through seven reductions on 'class strings'; each reduction is accompanied by an injective forward map and an explicit inverse walk-back procedure, and the final reduced problem is solved by explicit local solutions in §12.

Significance. If the proof is completed, the paper answers Kourovka Problem 20.23, improves the four-class result of Garonzi and Maróti to three classes, and complements the Lie-type results. The class-string machinery is original and substantial: the seven reductions are individually equipped with explicit inverse constructions, and the overall argument has no apparent circularity in the combinatorial core. The main weakness is the reliance of Proposition 3.2 on an unverified adaptation of a character-theoretic lemma; this proposition is load-bearing for the large-class half of Theorem 1.2. I was not able to certify every long cycle manipulation in §§5–12 by hand, but the structure of the reductions is coherent and the local solutions in §12 are directly checkable.

major comments (2)

[§3, Proposition 3.2] The split-class case of Proposition 3.2 is load-bearing for Theorem 1.2: it is the only mechanism that reaches classes C with |C| ≥ |Alt(n)|^{2δ/3}. In the case where both C1 and C2 are split classes, the proof states that a character-theoretic argument is 'essentially contained in [GM21, Lemma 5.3(i)]' with m ∈ {n−1, n}, and that 'the technique works out similarly' for m ∈ {n−3, n−2}. This is an unverified adaptation: changing m changes the size of the class O_m and can affect which inequalities or which dominant character are used. The constructive alternative offered in the same paragraph is circular, because it applies Theorem 1.4, the very statement that Proposition 3.2 is used to prove. As written, the large-class half of Theorem 1.2 is therefore unsupported unless the split-class case for m ∈ {n−3, n−2} is proved in full.
[§3, Proposition 3.2, first paragraph] The proof of Proposition 3.2 derives m ≥ k1 + k2 from the hypothesis |C1||C2| ≥ |Alt(n)|^{1+δ} by applying Proposition 2.3(a) to each class. But Proposition 2.3(a) requires |C_j| ≥ |Alt(n)|^{1−δ1} for the individual class, and the product hypothesis does not imply this: one factor may be much smaller than |Alt(n)|^{1−δ} while the other is large. In the application inside Theorem 1.2 the three classes are individually large, so the statement can be repaired by adding that assumption, but the proposition as stated is not proved by the given argument.

minor comments (5)

[§1.1] In the paragraph on special classes, 'the union of two 1 distinct classes' contains a stray '1'; it should read 'two distinct classes'.
[Abstract and §1.2] The claims that 'no character theory is involved' and that the proof is constructive are overstated: Proposition 3.2 relies on the character-theoretic [GM21, Lemma 5.3(i)] for its split-class case. The statement should be qualified to the combinatorial proof of Theorem 1.4, or Proposition 3.2 should be made elementary.
[Proof of Theorem 1.3] The sentence 'By [MP21, Thm. 1.3], the result holds for every non-abelian finite simple group G ≠ Alt(n)' should be clarified, since the introduction presents [MP21] as a Lie-type result; if the cited theorem covers sporadic groups as well, a brief explanation would help the reader.
[§4] The notation µ is used both for the map from class strings to classes and for the map from element strings to permutations; although the context makes the meaning clear, a sentence flagging the overloaded notation would prevent confusion in the long reduction sections.
[§3, proof of Theorem 1.2] In the small-class argument, the variable c3 in 'the class C′ ∋ c3c^{-1}' appears to mean γ3; renaming it would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.4 is proved by independent elementary reductions, and the self-citations are load-bearing tools, not restatements of the target.

full rationale

I traced the full derivation chain. Theorem 1.2 is obtained from Theorem 1.4 together with Proposition 3.2. Theorem 1.4 is then proved independently in Sections 4–12: Proposition 12.1 supplies an aligned solution for Reduction VII, and Lemmas 11.2–5.1 undo the seven reductions in sequence. None of these steps invokes Theorem 1.2, Proposition 3.2, or the conclusion C1C2C3 = G. The apparently risky sentence in Proposition 3.2 ('just use Propositions 2.2–2.3(b) to show that |C1|,|C2|,|Om| ≥ |Alt(n)|^{1−δ} and then apply Theorem 1.4') is a forward reference to a theorem proved later by a disjoint argument, so it does not create a logical cycle: the proof of Theorem 1.4 never uses Proposition 3.2. The only self-citation is Proposition 2.3 from [DMP24], a published quantitative lemma relating class size to number of cycles; it is a parameter-free external result whose assumptions do not include the target statement, so it is legitimate independent support rather than circularity. The split-class case of Proposition 3.2 is delegated to [GM21, Lemma 5.3(i)] with an adaptation claim; if that adaptation is mathematically unsound, that is a correctness gap or risk, not a reduction-by-construction. Similarly, the inference from |C1||C2| ≥ |Alt(n)|^{1+δ} to |C1|,|C2| ≥ |Alt(n)|^{1−δ} in the constructive aside appears unsupported, but again this is a potential error in an auxiliary remark, not a circular derivation. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work to force the choice, and no known result is repackaged as a new one. The central claim is self-contained against external benchmarks (Dvir's theorem, Garonzi–Maroti, Dona–Maroti–Pyber), and the main reduction chain does not assume its own conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No empirical parameters are fitted. The paper's constants δ and n0 are existential thresholds, not calibrated to data. The proof is a finite chain of reductions over external quantitative lemmas; none of these lemmas states the target result, so the circularity burden is low.

assumptions (4)

standard math Quantitative cycle-count/class-size relation (Prop 2.3, from [DMP24] and [GM21, Lemma 2.3]).
Invoked in Corollary 2.4 and throughout Sections 5-12 to pass from |C_i| ≥ |G|^(1-δ) to bounds on the number and lengths of cycles, and to show shortened classes remain large. It is the main external quantitative input.
standard math Dvir's criterion (Prop 3.1, [Dvi85, Thm 5.1(iii)]): if C1, C2 are classes in Sym(n) with at most k1, k2 cycles and m ≥ k1 + k2, then C1C2 contains the class of one m-cycle.
Used in Proposition 3.2 to cover large classes when proving Theorem 1.2 from Theorem 1.4. The paper calls it elementary and constructive but does not reproduce the proof.
standard math Exceptional split-class case in Proposition 3.2 is asserted from [GM21, Lemma 5.3(i)] with the technique 'similar', or alternatively from Theorem 1.4.
This is the only place in the route to Theorem 1.2 where an argument is not fully written. The constructive alternative using Theorem 1.4 removes dependence on character theory and is non-circular because Theorem 1.4 is proved independently.
standard math Lie-type product theorem [MP21, Thm 1.3] and [LST24, Thm 7.4] for classical groups: three large normal subsets of a finite simple group of Lie type have product G.
Used in the proof of Theorem 1.3 for G different from Alt(n); not proved in this paper.

invented entities (1)

Class-string and label framework (string triples, labels N, L, T, S, P, and alignment properties C1-C6)
purpose: Encodes cycle structure with ordered cycles and labels so that seven reversible reductions can be performed and solutions can be glued back.
This is a technical device introduced by the paper, not a postulate about the external world; it has no falsifiable handle outside the proof itself.

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Pith. "Pith review of Products of three conjugacy classes in the alternating group." pith.science (2026). https://pith.science/paper/YR67EQGO

@misc{pith2026250506012,
  author       = {Pith},
  title        = {Pith review of: Products of three conjugacy classes in the alternating group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YR67EQGO}},
  note         = {Machine review of arXiv:2505.06012}
}

abstract

We prove that for $\delta$ small, $n$ large, and any three conjugacy classes $C_{1},C_{2},C_{3}$ of $G=\mathrm{Alt}(n)$ of size at least $|G|^{1-\delta}$ we have $C_{1}C_{2}C_{3}=G$. The result provides a positive answer to Problem 20.23 of the Kourovka Notebook [KM22], improves theorems of Garonzi and Mar\'oti [GM21] (using $4$ classes) and Rodgers [Rod02] (using larger classes), complements the known result for $G$ a simple group of Lie type [MP21] [LST24] [FM25], and is tight in several senses. Furthermore, since no character theory is involved, the proof can be used in principle to build a constructive algorithm that, given $g\in G$, outputs $c_{i}\in C_{i}$ such that $c_{1}c_{2}c_{3}=g$.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

E. Bertram. Even permutations as a product of two conjugate cycles. J. Combin. Theory Ser. A , 12(3):368--380, 1972

work page 1972

[2] [2]

J. L. Brenner. Covering theorems for FINASIGS VIII - A lmost all conjugacy classes in A _ n have exponent 4 . J. Aust. Math. Soc. , 25:210--214, 1978

work page 1978

[3] [3]

D. Dona, M. W. Liebeck, and K. Rekv\'enyi. Involutions in finite simple groups as products of conjugates. Comm. Algebra , 52(9):3750--3761, 2024

work page 2024

[4] [4]

D. Dona, A. Mar\'oti, and L. Pyber. Growth of products of subsets in finite simple groups. Bull. Lond. Math. Soc. , 56(8):2704--2710, 2024

work page 2024

[5] [5]

Y. Dvir. Covering properties of permutation groups. In Z. Arad and M. Herzog, editors, Products of Conjugacy Classes in Groups , pages 197--221. Springer-Verlag, Berlin (Germany), 1985

work page 1985

[6] [6]

Fulton and J

W. Fulton and J. Harris. Representation theory: a first course , volume 129 of Graduate Texts in Mathematics . Springer, New York (USA), 2004

work page 2004

[7] [7]

Fumagalli and A

F. Fumagalli and A. Mar\'oti. On the G owers trick for classical simple groups. J. Pure Appl. Algebra , 229(1), 2025. Article no. 107833

work page 2025

[8] [8]

Garonzi and A

M. Garonzi and A. Mar\'oti. Alternating groups as products of four conjugacy classes. Arch. Math. (Basel) , 116(2):121--130, 2021

work page 2021