Finite groups with the minimal generating set exchange property

Andrea Lucchini; Patricia Medina Capilla

REVIEW 3 major objections 5 minor 15 references

Finite groups with the minimal generating set exchange property

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that every finite group with the minimal generating set exchange property is solvable, completing the classification of such groups.

desk verdict Finishes the MGSE classification with a genuinely new proof technique and new D(G) bounds; the main theorem is plausible but the referee should push on the compressed steps in Lemma 2.4 and Section 3. read the letter →

arxiv 2506.01638 v1 pith:EYCBDOFT submitted 2025-06-02 math.GR

Andrea Lucchini , Patricia Medina Capilla This is my paper

classification math.GR MSC 20F05

keywords minimalgeneratingsetexchangepropertyMGSEfinitegroupssetsmaximalsubgroupsmonolithicsolvableprimitive

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 settles the question of which finite groups have the minimal generating set exchange (MGSE) property: whenever two minimal generating lists of a group generate it, every position in one list can be filled by some element of the other list while keeping the group generated. The main theorem states that any finite group with this property is solvable, and since solvable MGSE groups had already been classified, this gives a complete classification. The same techniques yield a quantitative statement about maximal subgroups: for every finite group $G$ and every maximal subgroup $M$, there is a generating set of size $d(G)$ with at least $d(G) - 2$ elements lying in $M$, and with $d(G) - 1$ elements in many important cases. A direct corollary is that $D(G) \geq d(G) - 2$ for every finite group and $D(G) = d(G) - 1$ for every solvable group, with the paper proposing the conjecture that $D(G) = d(G) - 1$ always holds.

What carries the argument

The central object is a flexible subgroup $H$ of an almost simple group $K$ with socle $S$: a proper subgroup such that $HS = K$ and there is a prime $r$ with $|h|_r \neq |hs|_r$ for every $h \in H$ and some $s \in H \cap S$. The proof uses the fact that every almost simple $K$ contains such an $H$, pulls it back through the embedding $G \leq K_1 \wr S_n$ to a proper subgroup $\tilde H < G$, and then proves Lemma 2.4, a refinement of the generation theorem of [13]. Given generators $g_1, \ldots, g_d$ of $G$ modulo $N$, the lemma lets one replace $g_1$ and $g_2$ by products with elements of $N$ so that the set still generates $G$ while a controlled generator lands inside $\tilde H$. Its proof uses good pairs $(x, y)$: the first coordinates of certain powers of $xg_1$ and $yg_2$ generate $S$ and satisfy non-conjugacy conditions, forcing the intersection with $S^n$ to be a product of full diagonal subgroups over a fixed partition. The block containing a marked point is pushed into the first cycle of a permutation, and delicate conjugacy arguments then show the block must collapse to size one, giving the generation statement that drives the contradiction for non-solvable groups.

What would settle it

Search for a finite non-solvable group with the MGSE property. Because quotients inherit the property, the first counterexample would have to be a group with a single minimal normal subgroup equal to a direct power of a non-abelian simple group. The paper's Lemma 2.4 predicts that inside every such group, two members of any minimal generating set can be multiplied by elements of that socle so that one lands in a fixed proper subgroup, forcing a violation of exchange; a computer search through primitive groups could either exhibit such a counterexample or confirm the predicted obstruction.

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Extended reading notes

Core claim

On its own terms, the paper establishes that solvability is necessary for the MGSE property: if a finite group $G$ has the property, then $G$ is solvable (Theorem 1.2). Because the solvable groups with the property were already classified in [11], this completes the classification of all finite MGSE groups. The proof assumes $G$ is not solvable, passes to a monolithic quotient with non-abelian socle $N = S^n$, and shows that within each almost simple section one can pick a flexible subgroup $H$ and pull it back to a proper subgroup $\tilde H$ that covers $G/N$. Lemma 2.4 then shows that two generators of any minimal generating set can be multiplied by elements of $N$ so that one of them lands in $\tilde H$, which forces a violation of the exchange property. Independently, Theorem 1.3 and Corollary 1.4 prove that every maximal subgroup contains at least $d(G) - 2$ members of some minimal generating set, with the stronger bound $d(G) - 1$ for maximal subgroups of type 1 and type 3, for type 2 when $G/N$ is cyclic or $d(G) > d(G/M_G)$, and for all solvable groups.

Load-bearing premise

The proof that a non-solvable group cannot have the exchange property depends on the existence, in every almost simple group, of a proper subgroup $H$ that together with the simple socle generates the whole section and has a prime-order wiggle; without such an $H$ the constructed subgroup $\tilde H$ would not exist, and the contradiction ruling out non-solvable examples would not get started.

Editorial extensions

If this is right

A finite group has the MGSE property exactly when it is one of the solvable groups already listed in [11]: a non-cyclic nilpotent example is a finite $p$-group, and a non-nilpotent solvable example has $G/\Phi(G) \cong N^\delta \rtimes H$ with $H$ cyclic of prime order and $C_H(V) = 1$.
For every finite group $G$ and every maximal subgroup $M$, at least $d(G) - 2$ elements of some minimal generating set can be chosen inside $M$ (Theorem 1.3).
For maximal subgroups of type 1 or type 3, and for type 2 when $G/N$ is cyclic or $d(G) > d(G/M_G)$, one can force $d(G) - 1$ elements of a minimal generating set into $M$; in particular $D(G) = d(G) - 1$ for every finite solvable group.
The universal lower bound $D(G) \geq d(G) - 2$ holds, and no group is known for which the minimum equals $d(G) - 2$, which motivates the conjecture $D(G) = d(G) - 1$ for every finite group.
The MGSE property passes to quotients, so the complete classification is closed under taking quotients.

Reading between the lines

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

Beyond the paper: the conjecture $D(G) = d(G) - 1$ could be tested directly on primitive groups whose maximal subgroups have socle intersection equal to a diagonal subgroup or trivial, the two cases where the paper's method does not reach the stronger bound; finding a group where every maximal subgroup captures only $d(G) - 2$ elements would refute the conjecture without touching Theorem 1.2.
Beyond the paper: the exceptional non-flexible maximal subgroups detected inside $\operatorname{Aut}(A_6)$ are natural building blocks for almost simple sections where the $D_M(G) = d(G) - 1$ argument must switch from Lemma 2.4 to Corollary 2.7; deciding whether those sections can appear inside any MGSE group would sharpen the boundary of the classification.
Beyond the paper: if the conjecture holds, then for every finite group some maximal subgroup can host all but one member of a minimal generating set, so the obstruction to the stronger bound would be the presence of very special primitive quotients rather than the generic hardness of finding generators.

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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

Quick take: the paper proves the missing half of the MGSE classification — if a finite group satisfies the exchange property it must be solvable — and adds a new quantitative result about maximal subgroups. The main theorem was conjectured in earlier work by the first author, so the statement is not a surprise, but the proof is a genuine extension of the Lucchini–Menegazzo generation theorem, and the flexible subgroup idea is the right tool. The D(G) ≥ d(G)−2 theorem with the corollary D(G) = d(G)−1 for solvable groups is new and worth having on its own.

What I think is solid: the overall architecture is clear. You reduce to monolithic groups with non-abelian socle, use the flexible subgroup to build preimages, and derive the contradiction from MGSE. The external results are cited properly and the paper is honest about where it relies on [13] and [14]. The D(G) section is well organised by primitive type.

The soft spots are all in the technical middle. Lemma 2.4 is the load-bearing lemma, and the paper says several things are "easy to check" that are not easy. In Proposition 3.2 some inequalities are asserted without derivation — the bounds involving |Out(S)| and the semiregular action need a line of justification. Proposition 3.8 hands the O'Nan-Scott case split to "easy to check", and Proposition 3.9's Magma computation has no transcript. These are fixable in revision, but as written they make the proof harder to verify than it should be.

The deeper concern is the symmetric membership in Lemma 2.4. The paper promises two generating sets, one with v2g2 in tilde H and one with w1g1 in tilde H, and the proof of the symmetric case is compressed into a sentence about choosing g or h in [14, Lemma 7]. In the fixed-point-free part, the final contradiction step says we can alter some y-coordinate to replace b_i,q by a conjugate without changing anything else; that step is not shown. If that alteration does not preserve the good-pair conditions or keep x in tilde H, the contradiction is not established. I am not claiming the lemma is false — the authors are experts and the structure is plausible — but the paper asks the referee to supply a nontrivial amount of detail at exactly the point where the main theorem depends.

Who is this for: people working in finite group generation, especially generation of monolithic groups. The result is niche but it settles an open question, and the D(G) estimates are useful. It absolutely deserves referee time. My recommendation: send it out, but ask for a revised version with the compressed steps expanded.

Referee Report

3 major / 5 minor

Summary. The paper studies the minimal generating set exchange (MGSE) property for finite groups: whenever two minimal generating sets of size d(G) are given, any generator from one can be replaced by some generator from the other so that the set still generates G. The main result (Theorem 1.2) asserts that every finite group with the MGSE property is solvable, which, combined with the earlier classification of the solvable case in [11], gives a complete classification. The proof reduces to monolithic groups with non-abelian socle and depends on a technical lemma (Lemma 2.4) about finding generating sets whose elements can be chosen to lie in a prescribed flexible subgroup. The paper also proves lower bounds on the maximal number of elements of a maximal subgroup that can occur in a minimal generating set (Theorem 1.3 and Corollary 1.4).

Significance. This paper addresses a question posed by Cameron, Lakshmanan S and Ajith and, if correct, provides a complete and elegant answer: the MGSE property forces solvability. The companion results on the number of generators of a maximal subgroup, in particular the lower bound D(G) ≥ d(G)−2 for every finite group, are new and of independent interest. The paper is clearly organized and builds on substantial prior work, including the deep generation theorem of Lucchini and Menegazzo and the notion of flexible subgroups. The main reservation is that the proof of the central technical lemma contains unproved assertions in exactly the places where the argument is most delicate, and the proof of Theorem 1.2 relies on a stronger version of Lemma 2.4 than is actually stated.

major comments (3)

[Lemma 2.4, final paragraph (fixed-point-free case)] The step 'we can alter some y_{n_{i,j}} for certain j in order to replace b_{i,q} with a conjugate of itself whilst not altering any other b_{i,j}' is asserted without proof. No construction of the alteration is given, and it is not verified that the alteration preserves the good-pair conditions (1)-(2), keeps x in \tilde H, or leaves the uniquely determined subgroups D_B and D_{Bρ} unchanged. Since this is the only mechanism that forces |B|=1, the conclusion K = G in the fixed-point-free case is not rigorously established. This case is essential for the proof of Theorem 1.2.
[Proof of Theorem 1.2, paragraph beginning 'Thus, we can find v1, v2 ...'] The argument requires two generating sets with v2g2 in \tilde H and w1g1 in \tilde H simultaneously. However, Lemma 2.4's 'Moreover' only guarantees that the first adjusted element g1v1 lies in \tilde H. The symmetric membership is only sketched in the σ-fixed-point case ('choose either g or h to be contained in H ∩ S'); in the fixed-point-free case, which is the generic situation when d(G/N) > 1, no symmetric construction is supplied. Hence the MGSE exchange step is not justified by the lemma as stated.
[Proposition 3.2] Several crucial inequalities are stated without derivation: 'We can show this is always strictly less than |S|^{n/2}', 'if c > 1, then this contradicts the inequality above', and the case S = A6, n = 1 relies on an unspecified computational check. Since Proposition 3.2 underpins the type-3 case of Theorem 1.3, these gaps need to be filled with explicit arguments or a reproducible computation.

minor comments (5)

[Proposition 3.8] The claim that every maximal subgroup R of K not containing S is flexible is dismissed as 'easy to check' after invoking the O'Nan-Scott classification. This is a substantial verification; the authors should provide a case-by-case table or cite a source where the check is carried out.
[Proposition 3.9] The two non-flexible maximal subgroups of Aut(A6) are described only by their sizes and a set of element orders. The Magma computation should be made reproducible, for example by giving the code or a precise construction of the subgroups, since the later discussion relies on these examples.
[Definition 2.3] The definition of \tilde H is slightly confusing as written: the set {(α_1,...,α_n)σ ∈ G : α_i ∈ H} should presumably be interpreted in the image of the embedding φ; please clarify the notation.
[Throughout] There are several typographical slips, e.g., 'Guralnik' in reference [8] should be 'Guralnick', and in the proof of Theorem 1.2 the expression '⟨g1,...,gd⟩N = G' should more precisely state that the image of this subgroup in G/N equals G/N.
[Lemma 2.4] The statement uses the conditions 'g1π has a fixed point' and 'g2π has a fixed point' without explicitly introducing the notation g_iπ in the preamble; the reader would benefit from a sentence defining π as the natural projection to S_n.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem 1.2 is derived from prior external lemmas, not from the MGSE property itself.

full rationale

The derivation does not reduce to its inputs by construction. The MGSE property (Definition 1.1) is the target notion, while the central tool, Lemma 2.4, is an independent generation statement for monolithic groups with non-abelian socle; it is proved by adapting the external theorem in [13], not by assuming MGSE or Theorem 1.2. Theorem 1.2 uses Lemma 2.1 (inheritance by quotients, from [11]) and then constructs a proper subgroup H-tilde from a flexible subgroup whose existence is cited from [14, Lemma 6]. These cited results do not include the MGSE property or the solvability conclusion, so they are independent support rather than circular premises. The conjectured classification is genuinely completed here: [11] handled the solvable case, and this paper supplies the non-solvable contradiction. The skeptic's concern about the 'we can alter y' step in Lemma 2.4 and about asymmetric H-tilde membership is a potential correctness or rigor gap, not a circular reduction: no equation or definition makes the theorem true merely by restating its assumptions. Self-citation is present, but the cited lemmas are externally falsifiable mathematical results whose statements do not contain the target theorem, so per the review rules they do not raise the circularity score.

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

The paper is pure mathematics and fits no numerical parameters. It leans on a network of published theorems, several by the same authors; these are prior results that do not contain the MGSE classification, so they are recorded here as axioms rather than as circular inputs.

assumptions (5)

domain assumption Existence of flexible subgroups in every almost simple group (Lucchini and Nemmi, Lemma 6).
Used in the proof of Theorem 1.2 to choose J and build H-tilde; if absent, the lifting argument fails.
standard math Generation theorem for monolithic groups with non-abelian socle: for every set g_i with G = N <g_i> there exist n_i in N with G = <g_i n_i>.
Foundation for Lemma 2.4 and the lifting technique.
standard math Embedding theorem: a monolithic group with socle S^n embeds into K1 wreath S_n.
Used to define K1, the embedding phi, and to work inside the wreath product.
standard math Probabilistic bound |Lambda| >= (53/90) |N|^d for primitive groups with a unique minimal normal subgroup.
Used in Proposition 3.2 to control the set of good preimages.
standard math Aut(S)/S is solvable (Schreier conjecture, now a theorem).
Used in Lemma 2.4 to pass from projections containing S to the conclusion that K cap S^n has full diagonal factors.

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Cite this review

Pith. "Pith review of Finite groups with the minimal generating set exchange property." pith.science (2026). https://pith.science/paper/EYCBDOFT

@misc{pith2026250601638,
  author       = {Pith},
  title        = {Pith review of: Finite groups with the minimal generating set exchange property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EYCBDOFT}},
  note         = {Machine review of arXiv:2506.01638}
}

abstract

Let $d(G)$ be the smallest cardinality of a generating set of a finite group $G.$ We give a complete classification of the finite groups with the property that, whenever $ \langle x_1, \dots, x_{d(G)} \rangle = \langle y_1, \dots, y_{d(G)} \rangle = G$, for any $1 \leq i \leq d(G)$ there exists $1 \leq j \leq d(G)$ such that $\langle x_1, \dots, x_{i-1}, y_j, x_{i+1}, \dots, x_{d(G)} \rangle = G.$ We also prove that for every finite group $G$ and every maximal subgroup $M$ of $G$, there exists a generating set for $G$ of minimal size in which at least $d(G)-2$ elements belong to $M$. We conjecture that the stronger statement holds, that there exists a generating set of size $d(G)$ in which only one element does not belong to $M$, and we prove this conjecture for some suitable choices of $M$.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[13]

Lucchini and F

A. Lucchini and F. Menegazzo, Generators for finite groups with a unique minimal normal subgroup, Rend. Sem. Mat. Univ. Padova 98 (1997), 173–191

work page 1997
[14]

Lucchini and D

A. Lucchini and D. Nemmi, On the connectivity of the non-generating graph, Arch. Math. (Basel) 118 (2022), no. 6, 563–576

work page 2022
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Lucchini, Generating hypergraphs of finite groups, Ars Combin

A. Lucchini, Generating hypergraphs of finite groups, Ars Combin. 161 (2024), 17–28

work page 2024
[1]

Aschbacher and R

M. Aschbacher and R. Guralnick, On abelian quotients of primitive groups, Proc. Amer. Math. Soc. 107 (1989), no. 1, 89–95

work page 1989
[2]

Baddeley and A

R. Baddeley and A. Lucchini, On representing finite lattices as intervals in subgroup lattices of finite groups, J. Algebra 196 (1997) 1-100

work page 1997
[3]

Ballester-Bolinches and L

A. Ballester-Bolinches and L. Ezquerro, Classes of finite groups, Mathematics and Its Appli- cations (Springer), 584. Springer, Dordrecht, 2006

work page 2006
[4]

Burness, R

T. Burness, R. Guralnick and S. Harper, The spread of a finite group, Ann. of Math. (2) 193 (2021), no. 2, 619–687

work page 2021
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J Cameron, Aparna Lakshmanan S, Midhuna V Ajith, Hypergraphs defined on algebraic structures, Commun

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[6]

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E. Detomi and A. Lucchini, Probabilistic generation of finite groups with a unique minimal normal subgroup, J. Lond. Math. Soc. (2) 87 (2013), no. 3, 689–706

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Guralnik and W

R. Guralnik and W. Kantor, Probabilistic Generation of Finite Simple Groups, J. Algebra (2) 234 (2000) 743-792

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M. Quick, Probabilistic generation of wreath products of non-abelian finite simple groups, Comm. Algebra 32 (2004), no. 12, 4753-4768. FINITE GROUPS WITH THE MINIMAL GENERATING SET EXCHANGE PROPERTY 15 Andrea Lucchini. University of Padova (Italy), Dipartimento di Matematica “...

2004

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [13]

Lucchini and F

A. Lucchini and F. Menegazzo, Generators for finite groups with a unique minimal normal subgroup, Rend. Sem. Mat. Univ. Padova 98 (1997), 173–191

work page 1997

[2] [14]

Lucchini and D

A. Lucchini and D. Nemmi, On the connectivity of the non-generating graph, Arch. Math. (Basel) 118 (2022), no. 6, 563–576

work page 2022

[3] [11]

Lucchini, Generating hypergraphs of finite groups, Ars Combin

A. Lucchini, Generating hypergraphs of finite groups, Ars Combin. 161 (2024), 17–28

work page 2024

[4] [1]

Aschbacher and R

M. Aschbacher and R. Guralnick, On abelian quotients of primitive groups, Proc. Amer. Math. Soc. 107 (1989), no. 1, 89–95

work page 1989

[5] [2]

Baddeley and A

R. Baddeley and A. Lucchini, On representing finite lattices as intervals in subgroup lattices of finite groups, J. Algebra 196 (1997) 1-100

work page 1997

[6] [3]

Ballester-Bolinches and L

A. Ballester-Bolinches and L. Ezquerro, Classes of finite groups, Mathematics and Its Appli- cations (Springer), 584. Springer, Dordrecht, 2006

work page 2006

[7] [4]

Burness, R

T. Burness, R. Guralnick and S. Harper, The spread of a finite group, Ann. of Math. (2) 193 (2021), no. 2, 619–687

work page 2021

[8] [5]

J Cameron, Aparna Lakshmanan S, Midhuna V Ajith, Hypergraphs defined on algebraic structures, Commun

P. J Cameron, Aparna Lakshmanan S, Midhuna V Ajith, Hypergraphs defined on algebraic structures, Commun. Comb. Optim. to appear

work page