{"id":"188df8af-caef-43fc-9a67-4c23156d2d95","arxiv_id":"2608.03935","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper proves a structure theorem for finite groups generated invariably by two elements of distinct prime orders, and classifies the almost simple groups that are invariably generated by a 2-element and a 3-element: they are exactly PGL2(3^{2^b}). The result settles the (2,3) case of a question of Zalesskii and contributes to the Dolfi, Guralnick, Herzog and Praeger conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'only if' direction of Theorem 5.3 hinges on Lemma 7.4, whose complete list of pairs (K,d) with |C_K(d)| odd is taken from [9, Theorem 3] without restatement or independent derivation; an incomplete list would collapse the Lie-type exclusions in §§7.3–7.5.","rationale":"The reader's weakest assumption identifies the same load-bearing point: Lemma 7.4 depends on the completeness of an external classification from [9, Theorem 3], and the sporadic p-broad checks are delegated to Magma code not visible in the manuscript body. I agree with that assessment. I found no explicit mathematical contradiction in the paper itself; the structural theorems in Sections 2 and 8 are internally coherent, and the proof of Proposition 6.1 for the positive direction PGL2(3^{2^b}) is checkable in the text. The 'only if' direction, however, funnels through Lemma 7.4: after PSL2 and small-rank/special cases, every remaining Lie-type group is excluded using the parity conclusion plus p-broad arguments. Since that parity conclusion is not derived in the manuscript, the classification claim is only as strong as the cited theorem and the associated computational checks. This does not overturn the reader's conditional verdict; it reinforces it. The requested adjustment is therefore unchanged: the paper should be accepted only once the supplemental code is shipped and the reduction from [9, Theorem 3] is made explicit and independently checkable. No ad hominem or theatrical criticism is intended; this is a standard reproducibility concern for a proof that leans on a specialized external classification.","tokens_in":25846,"tokens_out":15516,"duration_ms":161964,"concrete_test":"Obtain the exact statement and proof of [9, Theorem 3] and independently re-derive the reduction in Lemma 7.4: for every pair (K,d) on the cited list, verify (a) K is either one of the claimed survivors or is excluded by Proposition 6.1 and Lemmas 5.4, 5.5, 7.2, 7.3, (b) |C_K(d)| is odd, and (c) d is conjugate into every parabolic subgroup of K. In parallel, run an independent Magma/GAP enumeration over all finite simple groups of Lie type of order at most 10^12 (in particular all such groups of rank at most 8) of order-3 elements d with |C_K(d)| odd and compare the resulting pairs with the list asserted by [9, Theorem 3]; any unlisted survivor would disprove Lemma 7.4 and hence Theorem 5.3. If the enumeration finds no counterexample in the tested range and the infinite-family derivation is checked independently by a second algebraist, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 7.4 asserts that |C_K(d)| is even for every remaining Lie-type socle, and the reductions in Lemmas 7.5–7.20 and the proof of Theorem 5.3 all rely on this parity fact. The lemma's proof is: assume |C_K(d)| odd, invoke [9, Theorem 3] as the list of possible pairs (K,d), exclude PSL2, alternating, sporadic, PSL3 and Ree groups, and claim that only PSL4(3^a), PSU4(3^a), and G2(3^a) survive, with d∈K and d conjugate into every parabolic subgroup. The completeness of that list, the reduction to the survivors, and the parabolic-subgroup property are all asserted but not demonstrated in the manuscript. If [9, Theorem 3] is incomplete, or if its hypotheses differ from the setting in which Lemma 7.4 applies, the parity conclusion fails and with it the exclusion of all Lie-type groups outside PGL2(3^{2^b}). This is an external-dependence concern rather than an observed internal contradiction: the surrounding argument is coherent, but its central claim is only as secure as that cited classification. The same audit-by-reputation pattern appears in Lemmas 3.4 and 3.6, where sporadic p-broad subgroup checks are deferred to Magma code in supplemental materials not present in the body. Lemma 7.10 additionally depends on [9, Theorem 1], further concentrating the proof on the reliability of [9].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finite groups invariably generated by two elements of distinct prime orders s and t. It introduces the almost semisimple residual A(G) and proves Theorem A, a normal-series structure theorem for such groups: either the group is a soluble {s,t}-group or there is a series with a soluble {s,t}-quotient, an almost semisimple quotient whose simple factors have order divisible by st, a nilpotent section of order coprime to st, and a Hall {s,t}-subgroup residual. Theorem B states that any non-abelian simple subgroup of an invariably (s,t)-generated group embeds in an almost simple quotient. The rest of the paper specializes to (s,t) = (2,3). The central technical theorem, Theorem 5.3, asserts that an almost simple group G is invariably (2,3)-generated if and only if G ≅ PGL2(3^{2^b}); this is then combined with Theorem A to yield Theorem C, which describes the normal structure of all non-soluble (2,3)-invariably generated groups. The proofs rely on the classification of finite simple groups, broad and p-broad subgroups, and extensive case analysis for groups of Lie type. Theorems A and B are argued without CFSG beyond the Schreier property.","tokens_in":26239,"tokens_out":11175,"duration_ms":111565,"significance":"If the classification in Theorem 5.3 is correct, the paper gives a clean and somewhat surprising answer: among almost simple groups, only PGL2(3^{2^b}) are invariably (2,3)-generated. The structural Theorem A is a valuable tool in its own right, and the paper explicitly constructs examples showing that the residual section can be nontrivial and that isomorphic direct factors are excluded. Theorem B and Corollary 1.6 give a negative answer to Zalesskii's question. The introduction of p-broad subgroups and the parity setup around Lemma 7.4 are likely to be useful. The main limitation is auditability: several load-bearing steps are delegated to the external reference [9] without restatement, and sporadic-group checks are delegated to Magma code in supplemental materials that is not present in the preprint body. These gaps do not by themselves show an error, but they make the proof difficult to verify as written.","major_comments":[{"comment":"The proof of Lemma 7.4, and hence the exclusion of all Lie-type groups outside PGL2(3^{2^b}), rests entirely on [9, Theorem 3] for the complete list of pairs (K,d) with |C_K(d)| odd. This theorem is not stated, its hypotheses are not checked against the present setting (K simple, d of order 3, d possibly an automorphism), and the reduction to the surviving cases PSL4(3^a), PSU4(3^a), G2(3^a) is asserted without demonstration. In particular, the claim that in these surviving cases d is contained in a conjugate of every parabolic subgroup is not justified in the text. Since the parity conclusion |C_K(d)| even is used in Lemma 7.5, Lemma 7.7, and subsequently throughout §7, an incomplete or misapplied list would collapse the central classification. Please restate the relevant theorem, verify its hypotheses, and either derive or explicitly cite the parabolic-subgroup property used here.","section":"§7.2, Lemma 7.4"},{"comment":"The exclusion of sporadic simple groups in Lemma 5.5 depends on the assertions that almost simple sporadic groups have broad subgroups (Lemma 3.4) and 3-broad subgroups (Lemma 3.6). For all cases except Aut(HN) and Fi24 in Lemma 3.4, and the two exceptional pairs in Lemma 3.6, the verification is delegated to 'magma code in the supplemental materials'. No code is included in the preprint body, and the referee cannot audit these computations. These are load-bearing for Theorem 5.3: if even one sporadic pair lacks the claimed p-broad subgroup, the argument in Lemma 5.5 fails. Please either include the code or provide explicit subgroup generators and a countable verification that each conjugacy class of involutions/elements of order 3 is represented.","section":"§3, Lemmas 3.4 and 3.6; used in §5, Lemma 5.5"},{"comment":"This lemma, which is key to the characteristic-2 classical-group exclusion, relies on [9, Theorem 1] for the assertion that d normalizes a non-trivial 2-subgroup of K⟨d⟩. The exact statement of the theorem and a check that its hypotheses apply to the automorphism d at this stage of the proof are not given. Because Lemma 7.10 is used to force K to be one of PSLn(2^a), PΩ+_{2n}(2^a), F4(2^a), or E6(2^a), and z to be a graph or graph-field automorphism, this is another external dependence on [9] that should be made explicit and verifiable.","section":"§7.2, Lemma 7.10"}],"minor_comments":[{"comment":"The title in the arXiv header reads 'INV ARIABLY GENERATED'; this should be corrected to 'INVARIABLY GENERATED'.","section":"Title/abstract"},{"comment":"The phrase 'where we were not patient enough' is informal for a journal article; it also obscures whether the Magma run was intentionally incomplete or abandoned. A precise statement of which cases were checked and which were handled manually would be clearer.","section":"§3, Lemma 3.4"},{"comment":"The proof of Lemma 5.4 says 'For n=5, we check that Sym(5) and Alt(5) are not invariably (2,3)-generated.' This is true but the check is not displayed; a one-line explanation would be useful.","section":"§5, Lemma 5.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is coherent and the central classification is plausible, but the audit trail is not complete. I would ask the editor to ensure that the Magma code in the supplemental materials is actually made available, and to have a Lie-type specialist verify the application of [9, Theorem 3] in Lemma 7.4 and [9, Theorem 1] in Lemma 7.10. These are localized, fixable gaps rather than observed internal errors, so I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this is a serious paper that proves a genuinely new structural theorem for invariable (s,t)-generation and a complete (2,3) classification. If you work on finite simple groups or generation, it is worth your time. The main proof is coherent, and I could not find an internal contradiction. But there are two places where the argument is less checkable than I'd like, and they are exactly the places the authors lean on to close the Lie-type cases.\n\nWhat is new: Theorem A's residual decomposition is a real step forward. It describes any invariably (s,t)-generated group as a soluble {s,t}-group inside an almost semisimple quotient with non-abelian simple factors of order divisible by st, and a nilpotent coprime section controlled by those factors. The proof is mostly independent of CFSG except for the Schreier property, which is a nice feature. Theorem B, that a simple subgroup of an invariably (s,t)-generated group embeds in an almost simple such group, is a clean reduction. Theorem C gives a sharp (2,3) result: almost simple examples are exactly PGL2(3^{2^b}), with a strong normal series description for all (2,3)-generated groups. The p-broad subgroup notion extends Guralnick-Robinson's broad subgroups and is used honestly as a tool, not as a slogan.\n\nWhere I have reservations: Lemma 7.4 is the hinge of the 'only if' direction for Lie-type socles. It asserts |C_K(d)| is even, and the proof is 'assume odd, then [9, Theorem 3] lists the possibilities, exclude some, and the survivors are PSL4(3^a), PSU4(3^a), G2(3^a)'. That list is not restated, and the step 'in each case d is contained in a conjugate of every parabolic subgroup' is asserted without demonstration. If the list in [9] has a gap or its hypotheses don't match the setting, the parity argument collapses. This is an external-dependence concern, not an observed contradiction; the surrounding argument is coherent. I still want a referee to check the applicability of [9] carefully.\n\nSecond, Lemmas 3.4 and 3.6 delegate several sporadic and small-rank checks to Magma 'code in the supplemental materials' that is not in the preprint body. I don't doubt the checks, but the paper should ship the code and a reproducibility recipe.\n\nThe paper is honest about what it depends on, and the self-citation to [17] is a published lemma, not a circular move. The main theorems are formal enough that I see no fitted parameter or forced conclusion.\n\nBottom line: this deserves a serious referee. If the referee confirms the dependence on [9] is legitimate and the Magma code is supplied, the classification should stand. I'd bring it to a group theory reading group and would cite it if I worked on invariable generation.\n\nRecommendation: send to peer review, with a request for the supplemental code and a restatement of the [9] list used in Lemma 7.4.","headline":"A substantial advance in invariable generation: the (2,3) classification is new and mostly convincing, but Lemma 7.4 leans on an opaque external list and the Magma checks need to ship.","tokens_in":26686,"tokens_out":2890,"would_cite":true,"duration_ms":27006,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20D05","20D06","20D08","20D60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Almost simple groups invariably generated by an element of order 2 and an element of order 3 are precisely the groups PGL2(3^{2^b}).","keywords":["invariable generation","finite groups","prime order elements","almost simple groups","almost semisimple residual","p-broad subgroups","(2,3)-generation","PGL2(3^{2^b})"],"falsifier":"Search the small almost simple groups of Lie type singled out in Lemma 7.4 — for example PSL4(3), PSU4(3), and G2(3) — for an invariable (2,3)-sequence by exhaustive conjugate enumeration; the paper's Theorem 5.3 says none exists, so a single success would overturn the classification.","tokens_in":25773,"feed_emoji":"🧩","tokens_out":10449,"duration_ms":100134,"temperature":0.7,"pith_summary":"Given two distinct primes s and t, a finite group is invariably (s,t)-generated when it contains an element of order s and an element of order t such that, no matter how each is conjugated inside the group, the two conjugates still generate the whole group. The paper gives a structural description of all finite groups with this property (Theorem A): such a group is either a soluble group built only from the primes s and t, or it is a controlled extension whose quotients are 'almost semisimple' — built from non-abelian simple groups whose orders are divisible by st — and whose remaining residual part is soluble. For the case s=2, t=3 the description becomes a complete classification (Theorem C): the only almost simple groups that are invariably (2,3)-generated are the groups PGL2(3^{2^b}) with b≥1, and every non-soluble invariably (2,3)-generated group is a double cover of a direct product of pairwise non-isomorphic PSL2(3^{2^{a_i}}) factors. This settles a previously open embedding question for the pair (2,3) in the negative, since the smallest simple group that cannot be embedded into any invariably (2,3)-generated group is PSL3(2).","feed_headline":"Only PGL2(3^{2^b}) is (2,3)-invariably generated","feed_subtitle":"Complete classification of almost simple groups that stay generated no matter how conjugates are chosen.","key_machinery":"The argument is carried by two objects. The first is the almost semisimple residual A(G), the intersection of all normal subgroups whose quotient is almost semisimple; Theorem A shows A(G) is soluble and its quotient is a direct product of the simple socles of the almost simple quotients of G. The second is the p-broad subgroup: an elementary abelian p-subgroup that meets every conjugacy class of elements of order p; for p=2 this is the broad subgroup of [12], and the paper extends it to odd primes. Lemma 3.13 is the pivot: if G has an s-broad subgroup and (σ,τ) is an invariable (s,t)-sequence, then the centralizer of τ has order coprime to s. This converts the generation condition into a ce","core_discovery":"The central claim is Theorem 5.3: if K is a non-abelian simple group and G is almost simple with socle K, then G is invariably generated by an involution and an element of order 3 exactly when G ≅ PGL2(3^{2^b}) for some b ≥ 1. The proof runs through all possibilities for K — alternating groups, sporadic groups, PSL2(r^a), and the remaining groups of Lie type — and eliminates every family except PGL2(3^{2^b}). Along the way the authors establish Theorem A and Theorem B, which show that any finite invariably (s,t)-generated group has a normal series whose factors are either soluble {s,t}-groups or almost semisimple quotients with simple socle factors divisible by st, and that any non-abelian s","pith_inferences":["The p-broad machinery used here is not tied to (2,3); one could attempt the same exclusion for other pairs (s,t), and the hard step would be determining which almost simple groups admit s- and t-broad subgroups.","A testable consequence of the structural theorem is that an invariably (s,t)-generated group has a very restricted normal structure; for pairs with an almost simple classification in hand, Theorem A converts the embedding question into a subgroup-containment problem in a finite list.","For (2,3), the negative embedding answer is sensitive to the prime pair: the same argument suggests the answer for (2,p) may depend on whether PGL2(p^{2^b}) or a similar family exists, so checking small odd primes computationally would be a natural next experiment."],"forward_implications":["Every non-soluble invariably (2,3)-generated group has a normal series G ≥ K1 > K2 = A(G) ≥ K3 ≥ 1 with |G/K1|=2, K1/K2 a direct product of pairwise non-isomorphic PSL2(3^{2^{a_i}}) with a1<...