{"id":"6bcaca26-15df-48f4-81d5-3fcb2f2cc1e8","arxiv_id":"2608.00703","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For distinct primes p and q, the alternating group A_n, for every sufficiently large n, does not embed into any finite group invariably generated by an order-p and an order-q element.","lead":"For any two different prime numbers, a large alternating group can never be placed inside a finite group that is invariably generated by one element of each prime order. This settles an open problem from the Kourovka Notebook, a standard list of unsolved questions in group theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.10's type-D graph-automorphism step asserts that a graph automorphism is a projective similitude; standard treatments realize such automorphisms as dualities of the polar space, leaving a gap in the exclusion of large-rank classical groups.","rationale":"The Reader's verdict ACCEPT identified Collins's index bound in Proposition 3.14 as the weakest assumption. That is a plausible external-dependency concern, but the more immediate load-bearing issue is internal: the proof that large-rank classical groups are excluded relies on Lemma 3.10, and the type-D case of that lemma makes an unproved assertion about the natural-module realization of the graph automorphism. If the assertion fails, then Proposition 3.12 does not cover all classical groups; the projective-degree bound d(p,q) in Lemma 3.13 then is not established, and the main theorem is not proved. The concern is not an ad hominem or a stylistic preference; it is a precise question about the action of an outer automorphism on the natural module. Because the rest of the proof is coherent and the issue is localized to one subcase, the appropriate response is not rejection but a conditional accept pending verification or a revised argument for the type-D graph case. A negative result here would mean the paper needs a substantial patch, but the theorem could still be true. Hence CONDITIONAL rather than UNCHANGED.","tokens_in":13371,"tokens_out":21842,"duration_ms":201063,"concrete_test":"Consult the standard automorphism theorem for classical groups (e.g., GLS98, Theorem 2.5.12, or Kleidman–Liebeck) and determine whether the graph automorphism of PΩ^+(2n,q), n at least 5, lies in PΓO^+(2n,q). If it does not, Lemma 3.10's 'projective similitude' assertion is false. Then check the consequence directly: for a given L with n > 3L, does an order-2 graph automorphism of D_n stabilize some totally singular L-subspace, or can it be S-conjugated into the normalizer of the standard parabolic stabilizing such a subspace? Exhibit such a subspace or find a counterexample; this decides whether Proposition 3.12 survives without modifying Lemma 3.10.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 3.12 excludes every classical group of sufficiently large Lie rank by producing a proper subgroup (the normalizer of a standard parabolic Y) that meets every conjugacy class of elements of order p and q. The crucial conjugacy lemma for graph automorphisms with trivial field part is Lemma 3.10. In the untwisted type D case, the proof asserts: 'Then α is induced on the natural module by a projective similitude of the defining quadratic form.' This is not the standard picture: for D_n with n at least 5, the graph automorphism of PΩ^+(2n,q) is induced by a duality/correlation of the associated polar space, not by a collineation given by a projective similitude. A projective similitude preserves the quadratic form up to a scalar and therefore preserves the two families of maximal totally singular subspaces, whereas the graph automorphism is precisely the automorphism that interchanges the two families. If this assertion is false, the subsequent appeal to Lemma 3.7, which requires order-r elements represented by similitudes, does not apply, and the claim that some S-conjugate of α lies in N_Aut(S)(Y) is unsupported. Lemma 3.10 feeds directly into Lemma 3.11 and then Proposition 3.12, so the uniform bound B(p,q) and Theorem 4.1 depend on this step. The paper gives no citation for the type-D assertion, and it is not a consequence of the general facts it cites. This is an internal gap in the proof as written, not merely a disagreement with consensus.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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}.","tokens_in":13578,"tokens_out":12156,"duration_ms":112941,"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":[{"comment":"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.","section":"Section 3.2, Lemma 3.10"}],"minor_comments":[{"comment":"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":"Section 3.2, Lemma 3.11"},{"comment":"There is a minor typographical issue: 'oncenis sufficiently large' should read 'once n is sufficiently large'.","section":"Section 1, Abstract"},{"comment":"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'.","section":"Section 3.2, Lemma 3.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically interesting and the overall strategy is plausible, but the type-D graph-automorphism issue identified in Lemma 3.10 is a genuine gap in a load-bearing part of the proof. I do not recommend rejection because the gap is localized and may be fixable, but the final verdict should depend on whether the authors can supply a correct argument or citation for the existence of an invariant non-maximal singular subspace for type-D graph automorphisms. The use of [Pit24] for classical groups should also be checked carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nWhere 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.\n\nThe 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.\n\nWho 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.","headline":"Promising negative answer to a Kourovka problem, but a gap in the type D classical step leaves the main theorem unproven as written.","tokens_in":14179,"tokens_out":9655,"would_cite":false,"duration_ms":90638,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20D60","20D05","20D06","20B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any two distinct primes, some finite group cannot be embedded into a group invariably generated by elements of those orders.","keywords":["invariable generation","alternating groups","prime-order elements","embedding problem","finite simple groups","wreath product","projective representations","Kourovka Notebook"],"falsifier":"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.","tokens_in":13085,"feed_emoji":"🧩","tokens_out":6764,"duration_ms":52612,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every prime pair has a group that resists invariable generation","feed_subtitle":"For fixed p and q, all large alternating groups A_n fail to embed into any group invariably generated by orders p and q.","key_machinery":"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.","core_discovery":"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].","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Theorem A: a finite subgroup of GL_D(k), modulo its largest normal ℓ-subgroup, has a subgroup of index at most (D+2)! with abelian and Lie-type components; this yields the bound n ≤ D+2 for alternating sections.","marker":"[Col08]"},{"why":"Provides the classification of finite simple groups and the list of exceptional isomorphisms A_5, A_6, A_8 with groups of Lie type, used in Lemma 3.13 and Proposition 3.14 to exclude alternating sections in the Lie-type component.","marker":"[Wil09]"},{"why":"Supplies the geometry of classical groups—Witt decomposition, transitivity on totally singular subspaces—used in Lemmas 3.6 and 3.12 to show parabolic subgroups meet all conjugacy classes of prime-order elements.","marker":"[Tay92]"},{"why":"Gives the standard description of automorphisms of finite groups of Lie type as products of inner, diagonal, field, and graph automorphisms, used in Lemma 3.11.","marker":"[GLS98]"},{"why":"Provides the primary decomposition of semisimple similitudes, used in Lemma 3.7 to find invariant totally singular subspaces for prime-order elements in classical groups.","marker":"[Wal63]"},{"why":"Supplies the conjugacy of field and graph-field automorphisms to standard automorphisms, used in Lemma 3.11 to conjugate arbitrary automorphisms into parabolic normalizers.","marker":"[Pit24]"}],"fun_headline_variants":["Large A_n never embeds in any invariable p-q generated group","For each prime pair, big alternating groups resist embedding","A_n for n large dodges all invariable p and q generations","Fixed p and q exclude all sufficiently large alternating groups","No large alternating group sits inside an invariable p-q group"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Large A_n never embeds in any invariable p-q generated group","For each prime pair, big alternating groups resist embedding","A_n for n large dodges all invariable p and q generations","Fixed p and q exclude all sufficiently large alternating groups","No large alternating group sits inside an invariable p-q group"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001806,"raw_usage":{"total_tokens":7076,"prompt_tokens":874,"completion_tokens":6202,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":6116}},"tokens_in":490,"tokens_out":6202,"duration_ms":36865,"temperature":1.0,"reasoning_tokens":6116,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:21:02.093932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}