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Thompson-like characterization of solubility for products of finite groups

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Pith's one-line read For a finite group written as a product of two subgroups, pairwise soluble generation is equivalent to the commutator of the factors lying in the largest soluble normal subgroup.

desk verdict A genuine unification with a sound Section 3 reduction, but the almost-simple case analysis is an admitted outline: the theorem deserves publication, conditional on the tables being checked. read the letter →

arxiv 1908.03347 v1 pith:JKBYGMDN submitted 2019-08-09 math.GR

P. Hauck , L. S. Kazarin , A. Martínez-Pastor , M. D. Pérez-Ramos This is my paper · ORCID
classification math.GR MSC 20D4020D10
keywords SolubilityProductsofsubgroupsTwo-generatedS-connectionAlmostsimplegroupsIndependentprimesSolubleradicalFinite
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 establishes an exact structural counterpart of the classical two-generated criterion for solubility in the setting of factorized finite groups. It proves that for a finite group $G=AB$ with subgroups $A$ and $B$, the pairwise condition that $\langle a,b\rangle$ is soluble for every $a\in A$ and $b\in B$ holds precisely when the commutator $[A,B]$ lies in the soluble radical $G^S$, the largest soluble normal subgroup of $G$. The same equivalence already follows from a much weaker local test: it suffices to check pairs where $a$ is a $p$-element, $b$ is a $q$-element, and $p\neq q$. This gives a local-global bridge of the kind that matters in finite group theory, and it implies, among other things, that a product of two soluble subgroups satisfying the pairwise condition is itself soluble.

What carries the argument

The central object is the soluble graph $\Gamma_{\mathrm{sol}}(N)$ of a non-abelian simple group $N$: its vertices are the primes dividing $|N|$, and two primes are adjacent exactly when $N$ contains a soluble subgroup whose order is divisible by their product. The proof's workhorse is the notion of an independent pair of primes, i.e. a non-edge of this graph. For every candidate factorization of an almost simple group, the paper uses an order comparison to force one specified prime into $|A\cap N|$ and another into $|B\cap N|$, and then uses maximal-subgroup structure, primitive prime divisors of $p^k-1$, and published subgroup data to show the two primes are independent; such a pair would contradict condition (2). The final contradiction for products with non-simple socle is obtained by projecting the factorization onto the simple direct factors and applying minimality.

What would settle it

Take the exceptional almost simple types exhibited in the paper ($P Sp_6(2)$ and $U_4(3)$ with an outer automorphism of order $2$), construct their stated factorizations $G=AB$, and compute whether some $p$-element of $A\cap N$ and $q$-element of $B\cap N$ with $p\neq q$ generate a non-soluble subgroup while condition (2) holds; if such a pair exists, the main theorem is false, and if the claimed independent primes are found, the theorem survives these hardest cases.

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Extended reading notes

Core claim

The main theorem states that the following three assertions about a finite group $G=AB$ are equivalent: (1) $A$ and $B$ are $\mathcal S$-connected, meaning $\langle a,b\rangle$ is soluble for all $a\in A$, $b\in B$; (2) for every pair of distinct primes $p,q$, the subgroup $\langle a,b\rangle$ is soluble whenever $a\in A$ is a $p$-element and $b\in B$ is a $q$-element; and (3) $[A,B]\leq G^S$, the soluble radical of $G$. The proof shows that a minimal counterexample to (2) implies (3) would have to be almost simple, with a non-abelian simple socle $N$ not contained in either factor, and then eliminates every possible factorization of such a group. The elimination is carried out by exhibiting, for each factorization, two primes that are independent with respect to $N$, one dividing $|A\cap N|$ and the other dividing $|B\cap N|$, where independence means that $N$ has no soluble subgroup of order divisible by their product. A separate theorem records that such independent primes occur for all almost simple factorizations except two explicit isomorphism types, with further small exceptions when independence is required with respect to the full automorphism group.

Load-bearing premise

The load-bearing assumption is that the list of possible ways an almost simple group can be written as a product of two subgroups is complete and correctly checked, including the exceptional finite cases; the paper says much of that detailed checking is omitted.

Editorial extensions

If this is right

  • If $A$ and $B$ are $\mathcal S$-connected in $G=AB$, then $A^S=A\cap G^S$ and $B^S=B\cap G^S$; in particular, if $A$ and $B$ are soluble, then $G$ is soluble.
  • In an almost simple group, no nontrivial factorization can satisfy the distinct-prime condition: one of the factors must be trivial.
  • The theorem implies the known characterization of the soluble radical by pairwise solubility: an element $x$ lies in $G^S$ exactly when $\langle x,y\rangle$ is soluble for every $y\in G$.
  • For deciding $\mathcal S$-connection, only pairs of elements of distinct prime order need to be tested, so the condition is finite and checkable from Sylow-like data rather than from all pairs.
  • For almost all almost simple factorizations, the proof produces independent primes in the soluble graph, with only finitely many exceptions explicitly listed.

Reading between the lines

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

  • Because condition (2) only compares elements of different prime order, a finite list of prime-by-prime checks would decide $\mathcal S$-connection for any concrete finite group, so the theorem gives a route to computer verification.
  • The same local-to-global shape suggests a family of testable conjectures: for other group classes with a well-behaved radical, connection of a product may be equivalent to $[A,B]$ lying in the corresponding radical, though the simple-group classification would need to be replaced by class-specific data.
  • The independent-primes theorem can be read as a reusable statement: for any factorization of an almost simple group outside the two exceptional types, a non-solubility certificate of the required kind is guaranteed to exist, which may simplify future arguments about products of almost simple groups.
  • A natural next step, suggested by the proof's own exceptional cases, would be a complete determination for alternating groups of which factorizations admit independent primes; the paper only gives partial results there.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proves a Thompson-type characterization for factorized finite groups. The Main Theorem states that for a finite group G=AB with subgroups A and B, the following are equivalent: (1) A and B are S-connected, i.e. every subgroup <a,b> with a in A and b in B is soluble; (2) for all distinct primes p and q, every p-element of A and every q-element of B generate a soluble subgroup; and (3) [A,B] is contained in the soluble radical G^S of G. The proof proceeds by a minimal-counterexample argument in Section 3, reducing the problem to almost simple groups; Section 2 then proves Theorem 1, which says that in an almost simple group N <= G = AB <= Aut(N) with AN=BN=G and with A,B satisfying condition (2), one must have A=G or B=G. Theorem 1 is established by a case analysis over the Liebeck–Praeger–Saxl classification of maximal factorizations, using Lemma 2 for p-divisibility of A∩N and B∩N, and using independence of primes with respect to the soluble graph. A byproduct, Theorem 2, records the existence of independent primes in such factorizations, with exceptions for PSp6(2), U4(3), and several small Lie-type groups.

Significance. If the main theorem is correct, it is a substantial contribution: it simultaneously generalizes Thompson's theorem on solubility of two-generated subgroups, Carocca's theorem on products of S-connected soluble subgroups, and the Guralnick–Kunyavski–Plotkin–Shalev criterion for membership in the soluble radical. The Section 3 reduction is elegant and mostly self-contained, and it makes clear why the almost-simple case is the only obstruction. The paper also gives a useful byproduct about independent primes in the soluble graph. However, the central almost-simple argument is not presented in a fully verifiable form: the authors explicitly state that detailed checking work is omitted, and many rows of Tables 1–14 are justified only by phrases such as 'handled exactly as a1)' or 'easily checked'. Because the Main Theorem is reduced exactly to Theorem 1, the correctness of every such row is load-bearing. The result is plausible and the structure is coherent, but as printed the proof is not independently checkable without redoing the classification data. In addition, one step in the Section 3 reduction, the assertion 'B ∩ M = 1' in step (vi), is not justified in the text.

major comments (2)
  1. [Section 2 (Strategies; Tables 1–14)] The proof of Theorem 1 is the load-bearing part of the paper: Section 3(iv) reduces the Main Theorem to it. Yet the paragraph before Lemma 2 states that 'usually detailed checking work and easy calculations are omitted', and many entries in Tables 1–14 are approved by 'handled exactly as a1)', 'easily checked', or by reference to [13]. Two concrete examples: in §2.2.2(m), for N=U4(3), the proof that N_N(⟨y⟩) is the only maximal soluble subgroup of N containing an element of order 7 is asserted but not derived, and this uniqueness is essential for the contradiction; in §2.2.3(g), for N=PSp6(2), the claim that the only maximal soluble subgroups of N whose order is divisible by 15 are the normalizers of elements of order 5 is also asserted without proof. Since the minimal-counterexample reduction forces the almost-simple configuration, a single incorrect independence claim or a single missed factorization row would invalidate the Main Theorem. The manuscript should include the full verification for every row, or provide a reproducible machine-checkable supplement (for example explicit GAP or Magma checks against the ATLAS and the tables of [10] and [34]).
  2. [Section 3, step (vi)] The assertion 'B ∩ M = 1' in the first paragraph of step (vi) is not justified in the text. From step (v) one only knows that there is some h with V_i,A=V_i,G and V_i,B=1 for i≤h, and V_i,B=V_i,G and V_i,A=1 for i>h. If h<k, elements of B∩M with nontrivial coordinates in positions h+1,...,k are not excluded, so B∩M=1 does not follow as written. To make the step valid one must first prove that h is either 0 or k: because G=AN and N acts trivially on the components, A acts transitively on {L_1,...,L_k}, and a full projection in one coordinate for A forces full projections in every coordinate; similarly for B. After swapping A and B if necessary, one may therefore assume h=k, and then B∩M=1 follows from V_i,B=1 for all i. As printed, this is a gap in a load-bearing step, although it is repairable by adding this argument.
minor comments (5)
  1. [Section 2.2 (introductory paragraphs)] The word 'explicitely' is misspelled twice; it should be 'explicitly'.
  2. [Section 2.1 (Proposition 1)] There is a typo 'subgoup' for 'subgroup'.
  3. [Section 2.2.2(m)] The sentence '7 divides |A ∩ N | and 3 4 divides |B ∩ N |' should read '3^4 divides |B ∩ N |'; the superscript appears to be missing.
  4. [Section 2.2.3(b3)] The phrase 'whose order is divisible by 36, but not by 7' should read 'divisible by 3^6' rather than by the integer 36; as printed it is ambiguous.
  5. [Section 3, step (vi)] The notation 'V_{1,G} × ··· × V_{k,G}' is used as though it were a subgroup of M; a sentence explaining that this denotes the direct product in M = M_1 × ··· × M_k would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalence is proved from independent classifications and external radical-membership results; author self-citations are not load-bearing.

full rationale

The Main Theorem's equivalence is not built from its own conclusion. The implication (3) implies (1) is a direct one-line argument: if [A,B] is contained in the soluble radical G^S, then every generated subgroup <a,b> has derived subgroup in G^S and hence is soluble. The nontrivial implication (2) implies (3) is obtained by a minimal-counterexample reduction that eventually invokes Theorem 1 for almost simple groups. Theorem 1 is a case analysis over the Liebeck-Praeger-Saxl maximal factorization classification [34], the maximal subgroup data of [10], and the ATLAS [13], not over the Main Theorem or over a result equivalent to it. The lemmas imported from [2] (Lemmas 6-9) are external published criteria on maximal soluble subgroups; although one author of the present paper is a coauthor of [2], those results are stated with their own assumptions and do not presuppose the present theorem, so they are independent evidence rather than a self-citation chain. The only explicitly acknowledged gap is in Section 2, where the authors say 'usually detailed checking work and easy calculations are omitted' and several table entries are accepted as 'handled exactly as a1)' or 'easily checked'. That is a completeness and correctness risk for the case analysis, but it is not circularity: each table row is supposed to exhibit independent primes from external data, and no row is defined in terms of the theorem it is used to prove. No parameter is fitted and no prediction is renamed from an input. Accordingly, the paper's central claim has independent content and the circularity score is zero.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are present; the theorem is deterministic. The proof rests on accepted external classifications and prior radical criteria rather than on new postulates. The main epistemic risk is the unstated case-checking in Tables 1 through 14, not the introduction of new entities.

assumptions (8)
  • standard math The Liebeck-Praeger-Saxl classification of maximal factorizations of finite simple groups and their automorphism groups, including exceptional Lie type factorizations ([34], [29]).
    Theorem 1 in Section 2 enumerates all possible factorizations G=~A~B from this classification. A missed case would break the almost-simple analysis.
  • standard math The ATLAS [13] and the maximal subgroup tables of [10] for low-dimensional classical and small simple groups.
    Used in Section 2.4 and in exceptional classical cases to assert independence of primes and the list of maximal subgroups.
  • standard math Criteria for membership in the soluble radical via p-elements by Guest-Levy [25] and Dolfi-Guralnick-Herzog-Praeger [16, Theorem B].
    Main Theorem steps (ii) and (iii) conclude that certain elements lie in G^S or that G/N is soluble from solubility of two-generated subgroups.
  • standard math Baer-Suzuki type nonsolvable generation results of Gordeev-Grunewald-Kunyavski-Plotkin [22-24].
    Step (ii) uses the existence of conjugate elements generating a nonsoluble subgroup in a nonabelian simple group L.
  • standard math Malle-Saxl-Weigel generation theorem: in a nonabelian simple group L not isomorphic to U3(3), there are three involutions generating a nonsoluble subgroup ([36], Theorem A).
    Used in step (ii) for the case where the unique minimal normal subgroup is a direct product of several copies of L.
  • standard math Zsigmondy's theorem on primitive prime divisors, and Dickson's list of subgroups of L2(q) as presented in Huppert [31].
    Used throughout Section 2.2 to define the primes r, s, t and to prove independence for L2(q).
  • standard math Ramanujan-Bertrand prime distribution and Huppert's classification of soluble 2-transitive permutation groups.
    Used in Lemmas 4 and 5 for the alternating group case.
  • standard math Dixon's bound: a soluble subgroup of S_k has order at most 3^{k-1}.
    Step (vii) bounds |B| after showing B cap M = 1.

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Pith. "Pith review of Thompson-like characterization of solubility for products of finite groups." pith.science (2026). https://pith.science/paper/JKBYGMDN

@misc{pith2026190803347,
  author       = {Pith},
  title        = {Pith review of: Thompson-like characterization of solubility for products of finite groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JKBYGMDN}},
  note         = {Machine review of arXiv:1908.03347}
}
abstract

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, $n$-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups $G = AB$ with subgroups $A,\ B$ such that $\langle a, b\rangle$ is soluble for all $a \in A$ and $b \in B$. In this case, the group $G$ is said to be an $\cal S$-connected product of the subgroups $A$ and $B$ for the class $\cal S$ of all finite soluble groups. Our main theorem states that $G = AB$ is $\cal S$-connected if and only if $[A,B]$ is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.

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