The structure of finite groups invariably generated by two elements of prime order

Chris Parker; Inna Capdeboscq

REVIEW 3 major objections 3 minor 19 references

The structure of finite groups invariably generated by two elements of prime order

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read 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}).

desk verdict 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. read the letter →

arxiv 2608.03935 v1 pith:XD7FIKVN submitted 2026-08-04 math.GR

Inna Capdeboscq , Chris Parker This is my paper · ORCID

classification math.GR MSC 20D0520D0620D0820D60

keywords invariablegenerationfinitegroupsprimeorderelementsalmostsimplesemisimpleresidualp-broadsubgroups(23)-generationPGL2(3^{2^b})

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

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).

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

The classification stands on the completeness of a cited list of elements of order 3 with odd-order centralizers in Lie-type groups, plus Magma-audited checks for sporadic groups; if either is incomplete, the exclusion argument in Theorem 5.3 collapses.

Editorial extensions

If this is right

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<...<an, K2/K3 nilpotent of odd order with primes only from π(3^{2^{a_n}}−1)∪π(3^{2^{a_n}}+1), and K3 a Hall {2,3}-subgroup of K2.
A non-abelian simple group embeds in an invariably (2,3)-generated group if and only if it is isomorphic to PSL2(3^{2^a}) for some a≥1 or to Alt(5).
The embedding question for the pair (2,3) has a negative answer: PSL3(2), the smallest simple group not on that list, cannot be embedded into any invariably (2,3)-generated group.
For any primes s<t, Theorem B reduces the question of which simple groups appear inside invariably (s,t)-generated groups to the almost simple case, so future classifications can focus on almost simple groups.
Theorem A holds independently of the classification of finite simple groups apart from the Schreier property, so the structural description is available for all prime pairs.

Reading between the lines

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

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.

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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

Dear colleague,

The 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.

What 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.

Where 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.

Second, 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.

The 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.

Bottom 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.

Recommendation: send to peer review, with a request for the supplemental code and a restatement of the [9] list used in Lemma 7.4.

Referee Report

3 major / 3 minor

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.

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 (3)

[§7.2, Lemma 7.4] 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.
[§3, Lemmas 3.4 and 3.6; used in §5, Lemma 5.5] 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.
[§7.2, Lemma 7.10] 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.

minor comments (3)

[Title/abstract] The title in the arXiv header reads 'INV ARIABLY GENERATED'; this should be corrected to 'INVARIABLY GENERATED'.
[§3, Lemma 3.4] 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.
[§5, Lemma 5.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a case analysis relying on external classification results and prior lemmas, not on the theorem being proved.

full rationale

The paper's main results derive from the definition of invariable generation and standard group-theoretic reductions. Theorem A follows from elementary lemmas about quotients, residuals, Hall subgroups, and the Schreier property; Theorem B is a short quotient argument; Theorem 5.3 excludes all almost simple candidates except PGL2(3^{2^b}), with the positive direction explicitly constructed in Proposition 6.1. No fitted parameter, normalization, or quantity defined in terms of the conclusion is present. The load-bearing external dependencies are classification results, notably [9, Theorem 3] listing pairs (K,d) with |C_K(d)| odd, [11] for Lie-type structure, and [17, Lemma 4.8] (by one of the present authors) asserting that an automorphism normalizing a nontrivial p-subgroup normalizes a parabolic subgroup. These are independent published results with stated assumptions that do not include the target theorem; they are not restatements of PGL2(3^{2^b}) being invariably (2,3)-generated. The Magma checks for sporadic groups in Lemmas 3.4 and 3.6 are computational verifications, not hidden inputs equivalent to the conclusion. Therefore no step reduces, by construction or by self-citation, to the claim being proved.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

There are no numerically fitted parameters and no new physical or metaphysical entities. The 'almost semisimple residual' A(G) and p-broad subgroups are formal definitions internal to the proof, not unexplained external postulates.

assumptions (6)

domain assumption Classification of finite simple groups (CFSG), including GLS tables [11] for Lie-type structure, centralizers and conjugacy classes.
Used throughout Sections 5-7 to enumerate possible socles and automorphism classes; if CFSG or GLS tables are incomplete, the exclusion of Lie-type groups in Theorem 5.3 could fail.
standard math Schreier property: outer automorphism groups of finite simple groups are soluble.
Invoked in Lemma 2.5 and Theorem A to make G/K1 soluble and to justify the intersection structure of the almost semisimple residual.
domain assumption Guralnick-Robinson theorem [12, Theorem 1]: every quasisimple group has a broad subgroup.
Used in Lemma 3.3 and Lemma 7.5 to force z outside K once C_K(d) has even order.
domain assumption Aschbacher-Seitz [2] classification of involutions in even-characteristic Chevalley groups.
Used in Proposition 4.6 and Lemma 7.10 to identify which involutions have 3-divisible centralizers and which cases can occur.
domain assumption Gerhardt's list [9, Theorem 3] of pairs (K,d) with |C_K(d)| odd.
Load-bearing in Lemma 7.4 to conclude that |C_K(d)| is even for all remaining Lie-type socles; a missing case would break the central exclusion argument.
domain assumption Magma computations in supplemental materials for sporadic and small Lie-type p-broad subgroup checks.
Used to rule out sporadic groups in Lemmas 3.4 and 3.6 and to construct examples 1.3 and 1.5; the code is not present in the manuscript body.

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Pith. "Pith review of The structure of finite groups invariably generated by two elements of prime order." pith.science (2026). https://pith.science/paper/XD7FIKVN

@misc{pith2026260803935,
  author       = {Pith},
  title        = {Pith review of: The structure of finite groups invariably generated by two elements of prime order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XD7FIKVN}},
  note         = {Machine review of arXiv:2608.03935}
}

abstract

A group $G$ is invariably generated by two elements $a$ and $b$ if $G=\langle a^g,b^h\rangle $ for all $g,h\in G$. This paper provides a structural description of finite groups invariably generated by two elements of distinct prime orders $s$ and $t$. We illustrate the use of our main result in a case study with $s=2$ and $t=3$. In particular we prove that an almost simple group is invariably generated by an element of order $2$ and an element of order $3$ if and only if it is isomorphic to $\mathrm{PGL}_2(3^{2^b})$ for some $b \ge 1$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [9]

Gerhardt

S. Gerhardt. Order three normalizers of 2-groups.Comm. Algebra 45.8 (2017), pp. 3594–3605

work page 2017

[2] [17]

Parker and J

C. Parker and J. Saunders. Expansion of normal subsets of odd- order elements in finite groups.J. Lond. Math. Soc. (2)113.4 (2026), Paper No. e70534, 37

work page 2026

[3] [1]

Aschbacher and R

M. Aschbacher and R. Guralnick. Some applications of the first cohomology group.J. Algebra90.2 (1984), pp. 446–460

work page 1984

[4] [2]

Aschbacher and G

M. Aschbacher and G. M. Seitz. Involutions in Chevalley groups over fields of even order.Nagoya Math. J.63 (1976), pp. 1–91

work page 1976

[5] [3]

The Magma algebra system. I. The user language

W. Bosma, J. Cannon, and C. Playoust. “The Magma algebra system. I. The user language”. Vol. 24. 3-4. Computational alge- bra and number theory (London, 1993). 1997, pp. 235–265

work page 1993

[6] [4]

T. C. Burness and M. Giudici.Classical groups, derangements and primes. Vol. 25. Australian Mathematical Society Lecture Series. Cambridge University Press, Cambridge, 2016

work page 2016

[7] [5]

Detomi and A

E. Detomi and A. Lucchini. Invariable generation with elements of coprime prime-power orders.Journal of Algebra423 (2015), pp. 683–701

work page 2015

[8] [6]

Detomi, A

E. Detomi, A. Lucchini, and C. M. Roney-Dougal. Coprime in- variable generation and minimal-exponent groups.J. Pure Appl. Algebra219.8 (2015), pp. 3453–3465

work page 2015