Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements

Michael Ruofan Zeng; Ting Gong; Yong Yang

REVIEW 1 major objections 3 minor 1 cited by

Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements

T0 review · 1 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For any two distinct primes, some finite group cannot be embedded into a group invariably generated by elements of those orders.

desk verdict Promising negative answer to a Kourovka problem, but a gap in the type D classical step leaves the main theorem unproven as written. read the letter →

arxiv 2608.00703 v1 pith:YVPYVEQZ submitted 2026-08-01 math.GR

Ting Gong , Yong Yang , Michael Ruofan Zeng This is my paper · ORCID

classification math.GR MSC 20D6020D0520D0620B35

keywords invariablegenerationalternatinggroupsprime-orderelementsembeddingproblemfinitesimplewreathproductprojectiverepresentationsKourovkaNotebook

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 settles a problem from the Kourovka Notebook by showing that invariable generation by two prime-order elements is not a universal embedding property. For any fixed distinct primes p and q, the authors prove that the alternating group A_n cannot be embedded into any finite group that is invariably generated by an element of order p and an element of order q, once n is sufficiently large in terms of p and q. In particular, there exists a finite group—indeed, an alternating group—that cannot be embedded into any group with such an invariable generating pair. The proof works by bounding the degree of any alternating section of a finite simple group that can appear in this context, using a wreath-product conjugacy argument and a modular analogue of Jordan's theorem.

What carries the argument

The load-bearing mechanism is the wreath-product conjugacy lemma (Lemma 3.1): if a subgroup P of an almost simple group A has the property that every element of order r in A is conjugate by an element of the socle into P, and P maps onto A/S, then in the wreath product A ≀ Sym(t) every element of order r is conjugate by S^t into P ≀ Sym(t). Applying this to P as the stabilizer of a pq-element subset of S_n (for alternating groups) or as the normalizer of a standard parabolic subgroup (for classical groups) yields a single proper subgroup of G that meets every conjugacy class of elements of order p or q, which by the proper-subgroup criterion (Lemma 2.2) kills invariable generation. The remaining step is Theorem A of [Col08], which bounds the size of a finite subgroup of GL_D(k) after quotienting by the largest normal ℓ-subgroup by (D+2)!, forcing n ≤ D+2 for any alternating section A_n.

What would settle it

Determine, for p=2 and q=3, the explicit value of B(2,3) from the paper's constants and then search for an embedding of A_n with n > B(2,3) into a finite group invariably generated by an involution and an element of order 3; a single such embedding would refute Theorem 4.1. Alternatively, test Proposition 3.14 directly: find a finite subgroup of GL_D(k) that has an alternating section A_n with n > D+2, which would contradict the bound used in the proof.

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Extended reading notes

Core claim

The central discovery is a uniform finiteness statement: for every pair of distinct primes p and q there is an integer B(p,q) such that if a finite group H is invariably generated by an element of order p and an element of order q, then no composition factor of H has a section isomorphic to A_n with n > B(p,q). Consequently A_{B(p,q)+1} (or any larger alternating group) cannot embed into such an H. The proof reduces the embedding problem to a statement about the almost simple quotients of H: a chief factor of H of the form S^t forces a quotient S^t ≤ G ≤ Aut(S) ≀ Sym(t) that is itself invariably generated by elements of orders p and q, and the paper shows that such a G cannot contain arbitrarily large alternating sections. Alternating groups of large degree are excluded because a fixed intransitive subgroup of S_n meets every conjugacy class of elements of order p or q; classical groups of large Lie rank are excluded by showing a parabolic subgroup meets every such class; and the remaining simple groups are handled by bounding the degree of a faithful projective representation and applying Theorem A of the cited reference [Col08].

Load-bearing premise

The whole uniform bound rests on Theorem A of [Col08]: any finite subgroup of GL_D(k), after removing its largest normal p-subgroup, has a subgroup of index at most (D+2)! with a very restricted structure; if that index bound were wrong or weaker, the restriction on alternating sections would fail.

Editorial extensions

If this is right

Problem 21.142 of the Kourovka Notebook now has a negative answer for every pair of distinct primes.
For fixed p and q, all sufficiently large alternating groups are universal obstructions: none embeds into a finite group invariably generated by elements of orders p and q.
The obstruction is effective: B(p,q) is computable in principle from explicit constants, though far from optimal.
Any finite group that has A_n (with n > B(p,q)) as a subgroup cannot itself be invariably generated by an element of order p and an element of order q.
The result suggests that invariable generation by elements of small prime order severely constrains the possible composition factors of any containing group.

Reading between the lines

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

The method likely extends to other fixed sets of element orders, replacing {p,q} by any finite set of primes, with the same two-step exclusion of alternating and classical groups and a subsequent bounded projective-degree argument.
The wreath-product conjugacy lemma could be applied to other generation properties where one wants to show a proper subgroup meets all conjugacy classes of a given order.
The bound B(p,q) is probably very far from sharp; explicit computations for small primes may give much smaller obstructions than the paper's worst-case constants.
Because the obstruction is an alternating group, the result also shows that being embeddable in an invariably generated group is not a property inherited by arbitrary subgroups: A_n fails it even though it embeds in many groups.

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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 main theorem is likely right in spirit, but as written it has a load-bearing gap in the type D graph-automorphism step that I believe is real.

What's genuinely new: the paper gives a negative answer to Kourovka Problem 21.142, with an explicit effective bound, and the architecture is smart: chief-factor reduction, a wreath-product conjugacy lemma, a clean intransitive-subgroup argument for alternating groups, and a Collins-theorem bound to finish. The early lemmas are proved thoroughly, and the disclosure of AI use is honest.

Where it's soft: Lemma 3.10, type D. The proof states that a graph automorphism with trivial field part is induced by a projective similitude of the quadratic form. For D_n with n even, a projective similitude preserves the two families of maximal totally singular subspaces; the graph automorphism interchanges them, so it cannot be a projective similitude. For n odd a non-square similitude can swap the families, but the paper makes a blanket claim with no citation. Since Lemma 3.7 only applies to similitudes, the invariant-subspace conclusion doesn't follow. This is the exact step that excludes large-rank classical groups in Proposition 3.12, and that feeds into B(p,q) and Theorem 4.1. So the central theorem isn't established as written.

The reader's report didn't catch this; the stress-test note did, and I think it's correct. The rest of the proof is coherent, and the heavy external dependencies (CFSG, Collins, Wall) are standard and not a mark against the paper. The type D issue may be repairable by a direct correlation argument, but it's not in the manuscript.

Who this is for: finite group theorists working on generation and simple groups. It deserves a serious referee, not a desk reject; send it out and ask for a fix of Lemma 3.10. I would not cite the main theorem until that's corrected.

Referee Report

1 major / 3 minor

Summary. The manuscript proves that for any two distinct primes p and q, there exists a finite group that does not embed into any finite group invariably generated by an element of order p and an element of order q. This answers Problem 21.142 of the Kourovka Notebook in the negative. The proof proceeds by assuming such an embedding exists, passing to a chief factor of the form S^t and then to a group G with S^t ≤ G ≤ Aut(S) ≀ Sym(t), and proving a uniform bound B(p,q) on the degree of any alternating section A_n of S. Large alternating groups are excluded by a wreath-product conjugacy lemma using a proper intransitive subgroup; large-rank classical groups are excluded by an invariant-subspace argument for prime-order similitudes; the remaining simple groups are handled through bounded-degree projective representations and Collins's modular analogue of Jordan's theorem. Choosing n_0 > B(p,q) then gives the counterexample A_{n_0}.

Significance. If the proof is completed, this is a substantial contribution: it resolves an open problem from the Kourovka Notebook and provides effective, if non-optimal, constants. The reduction from embedding in an arbitrary invariable-generating pair to sections of composition factors is clean and correct. The elementary parts of the paper, including the wreath-product conjugacy lemma (Lemma 3.1), the large-alternating-group argument (Proposition 3.3), and the application of Collins's theorem (Proposition 3.14), are proved in detail and appear sound. The main issue is a specific, load-bearing step in the treatment of type-D classical groups that needs repair before the result is established.

major comments (1)

[Section 3.2, Lemma 3.10] The type-D paragraph states that a graph automorphism of PΩ^+(2n,q) with trivial field part is induced on the natural module by a projective similitude of the defining quadratic form. This is not the standard picture: a projective similitude preserves the two families of maximal totally singular subspaces, whereas the graph automorphism of D_n interchanges them. Consequently Lemma 3.7, which applies only to elements represented by similitudes of the given form, cannot be invoked for this automorphism. The claim that some S-conjugate of α lies in N_Aut(S)(Y) for a nonterminal parabolic Y is therefore unsupported. Since Lemma 3.10 feeds directly into Lemma 3.11 and then Proposition 3.12, the exclusion of type-D classical groups of sufficiently large Lie rank is not established as written, and the uniform bound B(p,q) depends on this step. Please replace this paragraph with a correct argument or a precise reference showing that every such graph automorphism stabilizes a non-maximal totally singular L-space under the stated nonterminal hypothesis, or handle type-D graph automorphisms separately in Proposition 3.12.

minor comments (3)

[Section 3.2, Lemma 3.11] The paper cites [Pit24, Proposition 2.5(4)] for the statement that outer automorphisms with nontrivial field part are InnDiag(S)-conjugate to standard field or graph-field automorphisms. Since [Pit24] is a paper on exceptional groups, the authors should confirm that the cited proposition applies to finite classical groups as used here, or replace it with a standard reference such as [GLS98].
[Section 1, Abstract] There is a minor typographical issue: 'oncenis sufficiently large' should read 'once n is sufficiently large'.
[Section 3.2, Lemma 3.7] In the proof of Lemma 3.7, the phrase 'the integer L/d well-defined' is missing a verb; it should read 'the integer L/d is well-defined'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof derives a uniform bound from external classification and linear-representation results; the type-D concern in Lemma 3.10 is a possible gap, not a self-referential step.

full rationale

The derivation chain is self-contained in the relevant sense: the claimed obstruction A_{n0} is not used as an input. The proof starts from a hypothetical embedding A_{n0} <= H and shows that any nonabelian composition factor S of H with S^t <= G <= Aut(S) wreath Sym(t) would force alternating and classical factors to satisfy explicit bounds N_alt(p,q), R_cl(p,q), and d(p,q), none of which is fitted to the final A_{n0}. The bounds are produced by independent lemmas: the subset-stabilizer argument (Proposition 3.3) uses only elementary cycle structure; the classical-group exclusion (Proposition 3.12) uses Wall, Taylor, GLS, and Piterman; the linear-group bound (Proposition 3.14) uses Collins's Theorem A. No step quotes a self-citation as justification for a mathematical claim; the only self-citation, [GZY26], is the AI-usage disclosure and is not load-bearing. The reviewer-flagged concern about Lemma 3.10, that a D_n graph automorphism is not induced by a projective similitude, is a possible mathematical gap, not a circular reduction of the theorem to its assumptions. Under the stated rules, correctness gaps of that kind are out of scope for the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard group theory facts and several deep external theorems, all clearly cited. No free parameters are fitted to data and no new mathematical entities are postulated. The paper's own new lemmas are proved in full.

assumptions (5)

domain assumption Classification of finite simple groups (CFSG)
Used in Lemma 3.13 and Proposition 3.14 to enumerate the finite simple groups remaining after excluding large alternating and large-rank classical groups, and to bound their faithful projective representation degrees.
domain assumption Collins's Theorem A (modular Jordan theorem)
Proposition 3.14 relies on the bound [bar U:LN] <= (D+2)! for finite subgroups of GL_D(k) with D >= 71; this index bound is what forces alternating sections to be small.
domain assumption Automorphism structure theory of finite simple groups of Lie type (GLS98, Piterman 2024)
Lemma 3.11 uses the standard classification of automorphisms of classical groups and Piterman's Proposition 2.5(4) on conjugacy of field and graph-field automorphisms to standard forms.
standard math Wall's classification of semisimple similitudes (Wall 1963)
Lemma 3.7 relies on the isotypic decomposition of semisimple similitudes and the duality involution on irreducible factors of T^r - lambda.
standard math Transitivity of classical groups on flags and totally singular subspaces (Witt's extension theorem, Taylor 1992)
Propositions 3.10 and 3.12 use transitivity to move invariant subspaces to the fixed standard parabolic subgroup.

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Cite this review

Pith. "Pith review of Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements." pith.science (2026). https://pith.science/paper/YVPYVEQZ

@misc{pith2026260800703,
  author       = {Pith},
  title        = {Pith review of: Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YVPYVEQZ}},
  note         = {Machine review of arXiv:2608.00703}
}

abstract

For any distinct primes $p$ and $q$, we prove that there is a finite group which does not embed into any finite group invariably generated by an element of order $p$ and an element of order $q$. This gives a negative answer to Problem 21.142 of the Kourovka Notebook \cite{kourovka21}. In fact, for every fixed pair $p,q$, the group $A_n$ cannot embed into such a group once $n$ is sufficiently large in terms of $p$ and $q$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The structure of finite groups invariably generated by two elements of prime order
math.GR 2026-08 conditional novelty 7.0 of 10

The almost simple groups invariably generated by an element of order 2 and an element of order 3 are exactly PGL2(3^{2^b}) for b ≥ 1.

Reference graph

Works this paper leans on

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Pith tools

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

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and Mortimer, Brian , title =

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, title =

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Wall, G. E. , title =. Journal of the Australian Mathematical Society , volume =. 1963 , pages =

work page 1963

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Piterman, Kevin I. , title =. Algebra & Number Theory , volume =. 2024 , doi =. 2301.02570 , archivePrefix =

work page Pith review arXiv 2024

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On the diameter of intersection graphs of finite groups

Lee, Melissa and Rekv. On the diameter of intersection graphs of finite groups , year =. 2403.04157 , archivePrefix =

work page Pith review arXiv

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