Commuting graph of $A$-orbits

G\"ulin Ercan; \.Ismail \c{S}. G\"ulo\u{g}lu

REVIEW 4 major objections 5 minor 15 references

Commuting graph of $A$-orbits

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read If every prime-element orbit is adjacent to every other across distinct primes, the finite group is nilpotent.

desk verdict A solid, clearly written paper introducing a natural graph that unifies the commuting graph and the conjugacy-class commuting graph; the main nilpotency theorem is real, and the stress-test's worry about the Vasiliev–Vdovin citation does not hold up. read the letter →

arxiv 1908.06867 v2 pith:MCWXFDFX submitted 2019-08-19 math.GR

İsmail Ş. Güloğlu , Gülin Ercan This is my paper · ORCID

classification math.GR MSC 20D1020D1520D45

keywords commutinggraphA-orbitsgroupactionbyautomorphismsnilpotentgroupsFrobeniusprimetriangle-freefinite

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

This paper introduces the commuting graph of $A$-orbits $\Gamma(G,A)$: vertices are the orbits of a finite group $A$ acting by automorphisms on a finite group $G$, and two orbits form an edge when some element of one commutes with some element of the other. The main theorem states that if, for every pair of distinct primes $p$ and $q$, every $A$-orbit of a $p$-element is adjacent to every $A$-orbit of a $q$-element, then $G$ is nilpotent. A direct corollary is that whenever $\Gamma(G,A)$ is a complete graph for some automorphism group $A$, the group $G$ is nilpotent. The paper also shows that for solvable $G$, disconnection of $\Gamma(G,A)$ exactly detects Frobenius and 2-Frobenius structure, and it classifies the isolated-vertex and triangle-free cases. The upshot is a graph-theoretic commutation condition that forces a structural conclusion about the entire group.

What carries the argument

The central object is $\Gamma(G,A)$, the commuting graph of $A$-orbits: vertices are the orbits of $G\setminus\{1\}$ under the automorphism action of $A$, and two distinct orbits are joined when some element of one commutes with some element of the other. The argument connects this graph to the prime graph $\operatorname{GK}(G)$: partitioning the prime-order orbits by the prime $p$ gives a quotient graph isomorphic to $\operatorname{GK}(G)$. The nilpotency proof proceeds by minimal-counterexample induction, uses a unique minimal normal $A$-invariant subgroup, and invokes the classification of prime graphs of nonabelian simple groups as a black box to exclude the simple case; a module-decomposition argument then produces the final contradiction.

What would settle it

A direct counterexample would be a non-nilpotent finite group $G$ with an automorphism group $A$ such that for every pair of distinct primes $p,q$ and every $p$-element $x$ and $q$-element $y$, some $a \in A$ makes $x$ commute with $y^a$; searching small groups for such a pair would settle the point.

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Extended reading notes

Core claim

The central claim, Theorem 3.2, is that a finite group $G$ is nilpotent if the following holds: for any two distinct primes $p,q$ and any elements $x,y \neq 1$ with $x$ a $p$-element and $y$ a $q$-element, the orbits $xA$ and $yA$ are adjacent in $\Gamma(G,A)$, meaning some element of $xA$ commutes with some element of $yA$. The proof runs by minimal counterexample: every proper $A$-invariant subgroup and every quotient by a proper $A$-invariant normal subgroup is nilpotent, forcing a unique minimal normal $A$-invariant elementary abelian $p$-subgroup $M$ with $G/M$ nilpotent. It then rules out the case where $G$ is a nonabelian simple group by observing that the hypothesis would make the prime graph complete, contradicting the known classification of prime graphs of finite simple groups. The remaining nonsimple configuration is contradicted by decomposing $M$ into homogeneous components as a module and producing an element $x \in M$ whose centralizer is $M$, so its orbit cannot be adjacent to an orbit of a $q$-element. Theorem 3.3 draws the corollary: if $\Gamma(G,A)$ is complete for some $A \le \operatorname{Aut}(G)$, then $G$ is nilpotent.

Load-bearing premise

The load-bearing premise is the external classification input, used without proof, that no nonabelian finite simple group has a complete prime graph; if that input failed or was misapplied, the proof's exclusion of the simple case would collapse and Theorem 3.2 would be unsupported.

Editorial extensions

If this is right

If a single automorphism group $A$ makes $\Gamma(G,A)$ complete, $G$ is nilpotent; conversely, a nonabelian nilpotent group need not admit such an $A$, since $D_8$ with its full automorphism group does not give a complete graph.
For solvable $G$, $\Gamma(G,A)$ is disconnected exactly when $G$ is Frobenius or 2-Frobenius; when connected its diameter is at most 8, and when disconnected the number of components is one plus the number of $A$-orbits on the Frobenius complements.
A complete vertex $zA$ whose centralizer $C_G(z)$ is nilpotent forces $G$ to be nilpotent, and the same pattern with solvability instead of nilpotency also holds.
If $\Gamma(G,A)$ is triangle free, then $G$ is a CP-group; if it is nonsolvable, it must be one of $\operatorname{PSL}(2,q)$ for $q \in \{5,7,8,9\}$, $\operatorname{PSL}(3,4)$, or a group with a nontrivial normal 2-subgroup whose quotient is $\operatorname{PSL}(2,4)$ or $\operatorname{PSL}(2,8)$.
With $Z(G)=1$, an edgeless graph with more than one vertex occurs exactly when $G$ is $\operatorname{PSL}(2,5)$ or a Frobenius group with elementary abelian kernel and complement of prime order.

Reading between the lines

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

The criterion suggests a computational nilpotency test: for a given pair $(G,A)$, checking adjacency between orbits of elements of distinct prime orders is finite, and any non-nilpotent group passing the test would directly falsify Theorem 3.2.
Because $\Gamma(G,A)$ depends only on $A$ through its action, the theorem applies to any subgroup of automorphisms; one natural extension would be to replace automorphisms by arbitrary permutations of $G$ preserving the identity, though the proof's group-theoretic structure would not carry over automatically.
The proof inherits the full weight of the classification of simple-group prime graphs, so a classification-free proof would be needed if one wants the nilpotency result as an elementary theorem.
The connectedness theorem for solvable groups suggests that the number of components of $\Gamma(G,A)$ could serve as an invariant for recognising Frobenius and 2-Frobenius structure under arbitrary automorphism actions.

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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

Short version: this paper is worth a serious look. It introduces the commuting graph Γ(G,A) of A-orbits, which genuinely unifies the ordinary commuting graph (A trivial) and Herzog et al.'s conjugacy-class commuting graph (A = Inn G). The main theorem (3.2) is a real result: if orbits of p-elements and q-elements are always adjacent for distinct primes, then G is nilpotent. The companion solvability criterion (3.6) and the classifications of edge-less and triangle-free cases (4.2, 5.2) are also new and clearly presented. The proofs are mostly reductions to known classifications—Parker, Vasiliev–Vdovin, Delgado–Wu, Arad–Chillag—so the novelty is in the definitions and the structural statements rather than in new machinery. That is fine, and the paper gives helpful examples showing the limits of the theorems.

The stress-test concern about Theorem 3.2 Step (2) does not hold up. The phrase "no such simple group" refers to a simple group with complete prime graph, not to A_n with no prime in [n−3,n]. Corollary 7.6(2) of Vasiliev–Vdovin is the standard result that no finite simple group has a complete prime graph. The authors should restate that corollary explicitly, because the current wording is ambiguous enough to trip up a careful referee—but the cited claim is correct, and the central argument stands.

The real soft spots are smaller. Lemma 3.10's proof says "one can easily check" using primitive prime divisors; a referee will want that written out or replaced by a Zsigmondy citation. Theorem 4.2's exclusion of the non-PSL(2,5) CP-groups is also summarized as "one can observe," which is routine for experts but should be expanded. Theorem 2.2(ii) just cites Parker; that is acceptable. The paper leans on deep external classifications, but it says so and does not hide the reliance.

This paper is for people working on commuting graphs, prime graphs, or group actions and graph invariants. It is a good contribution, not a spectacular one. I would send it to a serious referee, expecting them to request clarifications and expanded routine checks, not a rewrite.

Referee Report

4 major / 5 minor

Summary. The paper introduces the commuting graph Γ(G,A) of A-orbits on a finite group G, whose vertices are the nontrivial A-orbits and whose edges join two orbits when they contain commuting representatives. The main results are: Theorem 3.2, asserting that if every orbit of a p-element is adjacent to every orbit of a q-element for all distinct primes p and q, then G is nilpotent; Theorem 3.3, that completeness of Γ(G,A) implies nilpotency; Theorem 3.6, a solvability analogue based on solvable generation of pairs; Proposition 3.11, a structural conclusion for complete vertices; Theorem 4.2, classifying when an edgeless Γ(G,A) with more than one vertex occurs; and Theorem 5.2, characterizing triangle-free Γ(G,A) for nonsolvable G in terms of known CP-groups. The proofs reduce largely to cited classifications of Frobenius and 2-Frobenius groups, prime graphs of simple groups, CP-groups, and fixed-point-free automorphisms.

Significance. If the main theorems are correct, the paper offers a useful new invariant: the quotient of the commuting graph by a group action, which interpolates between the ordinary commuting graph and the commuting graph of conjugacy classes. The theorems are unconditional group-structure statements, and the paper contains no fitted parameters or ad hoc assumptions; the arguments are deductions from published classifications. The organization is clear, and the examples in Remarks 3.5 and 3.8 are helpful. The central caveat is that several load-bearing steps are delegated to external results without precise statements, so the correctness of the main nilpotency and complete-vertex theorems cannot currently be checked from the manuscript alone.

major comments (4)

[Theorem 3.2, Step (2)] This step excludes the nonabelian simple case, so it is load-bearing for the main nilpotency theorem. The proof says that because GK(G) is complete, 2 is a complete vertex, and then cites [15, Theorem 7.1] and [15, Corollary 7.6(2)] to conclude that no such simple group exists. The exact statements of these results are not given, and the inference is ambiguous: the corollary must rule out simple groups with 2 as a complete vertex in the precise sense needed, not merely rule out some other class. The distinction is not cosmetic: for example A_27 has 2 as a complete vertex of its prime graph while GK(A_27) is not complete, so '2 is complete' and 'the prime graph is complete' are genuinely different conditions. Please restate the cited results and show explicitly that they imply the absence of a nonabelian simple group with complete prime graph; alternatively, give a self-contained proof of that fact.
[Lemma 3.10] The proof of Lemma 3.10 is not given: after reducing to field automorphisms of groups of Lie type, the text says 'Looking at the primitive prime divisors of these polynomials one can easily check that there exists a prime dividing |G| which does not divide |C_G(α)|.' This is the only proof of a statement used essentially in Step (5) of Proposition 3.11. Please provide the full check, or state and prove a general lemma with the relevant cyclotomic data; a citation to Table 6 of [3] together with a reference to a standard order formula would also need to be made explicit.
[Proposition 3.11, Step (6)] The final contradiction of Proposition 3.11 depends on two more unstated external facts: the classification consequence from [15] that a nonabelian simple group with complete vertex 2 is A_n with p=2, and the claim from [9, Corollary 5, Table 10.7] that if π(M)=π(H) with M simple and H=C_M(z), then the only possibility is M≅PSU(4,2), H≅S_6. The covering condition M=⋃_{a∈A} H^a is used to get π(M)=π(H), but the passage from this to the unique pair (M,H) is not demonstrated. Please state the relevant results precisely and indicate how they apply; otherwise the exclusion of the remaining case is not checkable.
[Theorem 4.2] The proof of Theorem 4.2 classifies the possibilities for G from the CP-group theorem and then excludes most of them with very brief remarks: 'one can observe that all nonsolvable groups other than PSL(2,5) ... do not satisfy the condition that each Sylow subgroup is elementary abelian', and a one-sentence exclusion of 2-Frobenius groups. Since Theorem 4.2 is the main result of Section 4, these exclusions should be written out, especially because the elementary-abelian-Sylow condition is exactly what distinguishes PSL(2,5) from the other candidate simple groups. The converse direction of the 'if and only if' is also only implicit.

minor comments (5)

[Theorem 3.2, Step (3)] The notation \bar G_p = P/M introduces P without defining it; please write 'where P is a Sylow p-subgroup of G containing M' to avoid ambiguity.
[Definition 1.1] The equality Γ(G,A)=Γ(G,A/C_A(G)) is clear but should be justified in a sentence, since the vertices are orbits of the action and the quotient acts with the same orbits.
[Throughout] The name 'Grünberg-Kegel graph' should be 'Gruenberg-Kegel graph' or 'prime graph' consistently, and the spelling should be checked in the abstract and Section 1.
[Theorem 4.2] The necessary condition on N_A(P)-orbits is stated in a separate sentence after the 'if and only if'; making it an explicit part of the equivalence would clarify the proof of the converse direction.
[Theorem 5.2] In the proof, the notation G=G/O_2(G) reuses the symbol G for the quotient; using \bar G would avoid confusion in the later paragraphs about Sz(8).

Circularity Check

0 steps flagged · score 0.0 of 10

Self-contained theorem-proving paper; no circularity found.

full rationale

The paper defines a new object Γ(G,A) and proves structural theorems about G under graph-theoretic hypotheses. The central claim, Theorem 3.2, is proved by induction with explicit reduction steps; the final contradiction uses the hypothesis directly, with no fitted parameters and no quantity defined in terms of the conclusion. The only potentially load-bearing external input is the cited prime-graph classification of Vasiliev and Vdovin ([15], Theorem 7.1 and Corollary 7.6(2)), used to exclude a nonabelian simple group with complete prime graph. That citation is to independent external work, not to the authors' own prior results, and the cited result does not itself assert nilpotency. Even if the application of [15] were too terse or vulnerable to misreading, that would be a correctness risk about external support, not circularity. The other cited results ([2], [5], [6], [7], [9], [13]) are likewise external benchmarks with explicitly stated assumptions. No step of the derivation reduces by construction to its own input, no fitted value is renamed a prediction, and no self-citation is load-bearing. The paper is therefore not circular.

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

The paper introduces a new graph but no new axiomatic entities. All free parameters are absent; the results are derived from published classification theorems. The listed axioms are the external results, mostly classifications, that the proofs invoke.

assumptions (8)

domain assumption Classification of finite simple groups (CFSG) and its corollaries used in the cited classifications.
Used implicitly throughout; the proofs invoke classifications that depend on CFSG, e.g. [5], [6], [9], [15].
domain assumption Vasiliev-Vdovin prime graph classification [15].
Used in Theorem 3.2 step (2) and Proposition 3.11 to rule out nonabelian simple groups with complete or almost-complete prime graphs.
domain assumption Parker's classification of solvable groups with disconnected commuting graph [13].
Used in Theorem 2.2 to conclude G is Frobenius or 2-Frobenius from disconnection of Γ(G,A).
domain assumption Delgado-Wu classification of groups in which every element has prime power order [5].
Used in Theorems 4.2 and 5.2 to enumerate possible nonsolvable groups.
domain assumption Arad-Chillag theorem on p-elements with subgroup centralizers [2].
Used in Proposition 4.1 to assert CG(g) is an elementary abelian CC-subgroup.
domain assumption Dolfi-Guralnick-Herzog-Praeger solvability criterion [6].
Used in Theorem 3.6 to identify pairs of primes generating nonsolvable subgroups.
domain assumption Glauberman-Guralnick-Lynd-Navarro result on Sylow centers and automorphisms [7].
Used in Proposition 3.11 for odd p to show the automorphism is inner.
domain assumption Liebeck-Praeger-Saxl results on primitive permutation groups [9].
Used in Proposition 3.11 to rule out the PSU(4,2) case.

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Cite this review

Pith. "Pith review of Commuting graph of $A$-orbits." pith.science (2026). https://pith.science/paper/MCWXFDFX

@misc{pith2026190806867,
  author       = {Pith},
  title        = {Pith review of: Commuting graph of $A$-orbits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MCWXFDFX}},
  note         = {Machine review of arXiv:1908.06867}
}

abstract

Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $\Gamma (G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on $\Gamma (G,A)$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[3]

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work page 1985
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Abdollahi, S

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Z.Arad, D.Chillag, On ﬁnite groups with conditions on th e centralizers of p-elements, J. Al- gebra 51 (1978) 164–172

work page 1978
[4]

M. R. Darafsheh, Groups with the same non-commuting grap h, Discrete Appl. Math. 157 (2009) 833–837

work page 2009
[5]

A. L. Delgado, Y. W u, On locally ﬁnite groups in which ever y element has prime power order, Illinois Journal of Mathematics 46 (2002) 885–891

work page 2002
[6]

S. Dolﬁ, R. M. Guralnick, M. Herzog, and C. E. Praeger, A ne w solvability criterion for ﬁnite groups, Journal of the London Mathematical Society 85 (2) (2012) 269–281

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Glauberman, R

G. Glauberman, R. Guralnick, J. Lynd, and G. Navarro, Cen ters of Sylow subgroups and automorphisms, Preprint. ArXiv.org

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R . A. Wilson, Atlas of ﬁnite groups, Clarendon Pres s, Oxford (1985)

work page 1985

[2] [15]

A. V. Vasiliev, E. P. Vdovin, An adjacency criterion for the prime graph of a ﬁnite simple group, Algebra and Logic 44 (6) (2005) 381–406. ˙Isma˙Il S ¸. G¨ulo˘glu, Department of Mathematics, Do ˘gus ¸ University, Istanbul, Turkey E-mail address : iguloglu@dogus.edu.tr G¨ul˙In Ercan, Department of Mathematics, Middle East echnical Un iversity, Ankara, Turk...

work page 2005

[3] [1]

Abdollahi, S

A. Abdollahi, S. Akbari, H. R. Maimani, Non-commuting gr aph of a group, J. Algebra 298 (2006) 468–492

work page 2006

[4] [2]

Al- gebra 51 (1978) 164–172

Z.Arad, D.Chillag, On ﬁnite groups with conditions on th e centralizers of p-elements, J. Al- gebra 51 (1978) 164–172

work page 1978

[5] [4]

M. R. Darafsheh, Groups with the same non-commuting grap h, Discrete Appl. Math. 157 (2009) 833–837

work page 2009

[6] [5]

A. L. Delgado, Y. W u, On locally ﬁnite groups in which ever y element has prime power order, Illinois Journal of Mathematics 46 (2002) 885–891

work page 2002

[7] [6]

S. Dolﬁ, R. M. Guralnick, M. Herzog, and C. E. Praeger, A ne w solvability criterion for ﬁnite groups, Journal of the London Mathematical Society 85 (2) (2012) 269–281

work page 2012

[8] [7]

Glauberman, R

G. Glauberman, R. Guralnick, J. Lynd, and G. Navarro, Cen ters of Sylow subgroups and automorphisms, Preprint. ArXiv.org

work page