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REVIEW 5 major objections 4 minor 15 references

Groups with maximum vertex degree commuting graphs

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Pith's one-line read The paper's main theorem gives a complete list of sixteen finite non-abelian groups with strong 5-star-free commuting graphs.

desk verdict Right answer, unfinished proof: the classification is plausible and the finiteness theorem is clean, but too many exclusions are dismissed with 'similar', and an off-by-one error leaves the |Z|=5 case open. read the letter →

arxiv 1908.08226 v2 pith:MDFBPY4S submitted 2019-08-22 math.GR

Sushil Bhunia , G. Arunkumar This is my paper · ORCID
classification math.GR MSC 20E99
keywords commutinggraphstrongstar-freeclaw-freestarnumbercentralizerclassequationfinitenon-abeliangroups
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 aims to settle which finite non-abelian groups have a commuting graph with no five-pointed star as a subgraph. The commuting graph of a group has one vertex for each non-central element, with an edge between two vertices exactly when the corresponding elements commute; 'strong 5-star-free' means the graph contains no copy of $K_{1,5}$. The main theorem claims a complete list of exactly sixteen such groups, ranging from $S_3$, $D_{10}$, $A_4$, $D_8$, $Q_8$, and $A_5$ to a handful of groups of orders 12, 16, and 24. If the classification is correct, it also yields the complete list of strong claw-free groups and shows that, for every $k$, only finitely many finite non-abelian groups are strong $k$-star-free. The point of the result is that a graph-theoretic forbidden-subgraph condition turns out to reduce a huge universe of groups to a short, explicit list.

What carries the argument

The core objects are the commuting graph $\Gamma(G)$, whose vertices are the non-central elements with edges between commuting pairs, and the centralizer-size criterion that detects stars. A $k$-star is the complete bipartite graph $K_{1,k}$, and 'strong $k$-star-free' means no such graph occurs as a subgraph. Corollary 2.2 converts the condition into the inequality $|C_G(x)|<k+1+|Z(G)|$ for every non-central $x$, so the classification reduces to bounding centralizer sizes. The class equation then carries the argument: it forces the possible orders of a group with a given pattern of centralizer sizes, and the surviving orders are checked against the known groups of those orders.

What would settle it

Calculate the class equations for every centralizer-size pattern in Equations (3.2) through (3.26); any pattern dismissed by Lemma 3.1 that yields an integer group order and an actual group with those centralizer sizes would refute Theorem 1.1. Independently, enumerate finite non-abelian groups of order up to 60, compute $|C_G(x)|-|Z(G)|$ for every non-central $x$, and check whether any group outside the theorem's list has all such values at most 5.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: a finite non-abelian group $G$ has strong $5$-star-free commuting graph $\Gamma(G)$ exactly when $G$ is one of $S_3$, $D_{10}$, $A_4$, $GA(1,5)$, $A_5$, $D_8$, $Q_8$, $D_{12}$, $C_4\rtimes C_3$, $SL(2,3)$, $(C_4\times C_2)\rtimes C_2$, $C_4\rtimes C_4$, $C_8\rtimes C_2$, $D_8\rtimes C_2$, $Q_8\rtimes C_2$, or $(C_4\times C_2)\rtimes C_2$, where $C_n$ is cyclic of order $n$ and $GA(1,5)$ is the affine group of the line over the field of five elements. In other words, every finite non-abelian group outside this list has some non-central element that commutes with at least five other non-central elements. The proof funnels the graph condition into a numerical one: by Corollary 2.2, $\Gamma(G)$ is strong $k$-star-free precisely when $|C_G(x)|<k+1+|Z(G)|$ for every non-central $x$. The class equation then restricts which patterns of centralizer sizes are possible, and each surviving pattern is identified with a specific group.

Load-bearing premise

The completeness of the sixteen-group list rests on the unshown claim that all the skipped arithmetic cases in Lemma 3.1 are impossible; the paper works two examples and says the rest are similar, and if any of those cases allows a group, the list is incomplete.

Editorial extensions

If this is right

  • Corollary 1.2: the same sixteen groups are the complete list of strong 4-star-free finite non-abelian groups.
  • Corollary 1.3: the only strong claw-free finite non-abelian groups are $S_3$, $A_4$, $D_8$, and $Q_8$; each has a commuting graph with no $K_{1,3}$ subgraph.
  • Theorem 1.4: for every natural number $k$, the collection $SF(k)$ of finite non-abelian groups whose commuting graph is strong $k$-star-free is finite.
  • The dihedral-group analysis shows the inclusions $SF(1)\subseteq SF(2)\subseteq\cdots$ are strict, so there are non-abelian groups with arbitrarily large strong star number.
  • Any finite non-abelian group not on the Theorem 1.1 list contains a $K_{1,5}$ subgraph in its commuting graph.

Reading between the lines

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

  • The authors do not classify ordinary induced claw-free commuting graphs, which require a vertex with three pairwise non-adjacent neighbours; because vertices within a centralizer often commute with each other, that list is likely to be larger than $S_3,A_4,D_8,Q_8$, and the centralizer data in this paper gives a direct way to compute it.
  • The same class-equation strategy should extend to strong $6$-star-free groups: add centralizer size 7 to the allowed trivial-center patterns and carry the non-trivial-center cases one step further; Theorem 1.4 guarantees the resulting list is still finite.
  • The finiteness phenomenon outruns the fixed-$k$ statement: any rule that restricts centralizer sizes to a finite set leaves only finitely many finite non-abelian groups, because the class equation has finitely many solutions once the denominators are drawn from a finite set.
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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

5 major / 4 minor

Summary. The paper studies the commuting graph of a finite non-abelian group, whose vertices are the non-central elements and whose edges join commuting distinct vertices. It defines a group to be strong k-star free when the star K_{1,k} is not a subgraph of this commuting graph, and claims a complete classification of finite non-abelian strong 5-star free groups (Theorem 1.1), with corollaries for strong 4-star free and strong claw-free groups. The proof strategy is to use Corollary 2.2 to bound centralizer sizes, then solve the class equation in cases according to the possible centralizer sizes, first with trivial center and then with nontrivial center. The paper also proves a finiteness theorem (Theorem 1.4) asserting that for each k there are only finitely many non-abelian groups whose commuting graph is strong k-star free.

Significance. If the classification is correct, it gives a complete and explicit answer to a natural graph-theoretic question about commuting graphs, and the finiteness theorem is a strong structural statement. The argument is not circular: it derives restrictions on centralizer sizes from the absence of a K_{1,5} subgraph and then uses the class equation, with no fitted parameters and no dependence on the paper's own claims. The external dependencies are mostly unstated classification facts about small groups, which is a gap in exposition but not a conceptual defect. However, several load-bearing proofs are sketched rather than completed, and one numerical bound is off by one, so the claimed completeness is not established as written.

major comments (5)
  1. [§3.1, Lemma 3.1] Lemma 3.1 rules out 20 of the 26 centralizer-size patterns for the trivial-center case, but the proof treats only Equations (3.2) and (3.11) and dismisses the rest with “The remaining cases are similar.” This is not a routine or harmless omission. For example, for Equation (3.6), where |C_G(x)| is 3 or 5, the class equation admits the integer solution n = 15 with one class of size 5 and three classes of size 3; eliminating this case requires a separate fact, such as the classification of groups of order 15, which is not supplied. Similar issues occur for (3.7), (3.9), and (3.10), whose class equations are solvable and are eliminated only by unstated small-group classifications. As written, this leaves open the possibility that one of the skipped cases contains a group not on the list in Theorem 1.1.
  2. [Lemmas 3.5 and 3.6] The proofs of Lemmas 3.5 and 3.6 say that the identification of GA(1,5) and A5 “follows from the class equation,” but a class equation alone does not determine a group up to isomorphism. For Lemma 3.6 the paper correctly states the A5 class equation 60 = 1 + 12 + 12 + 15 + 20, yet one still must prove that A5 is the only group of order 60 with trivial center and these centralizer sizes. Similarly, the class equation 20 = 1 + 4 + 5 + 5 + 5 does not by itself single out GA(1,5) among groups of order 20. This missing uniqueness/existence argument is needed for the completeness of Theorem 1.1.
  3. [§3.2, Lemma 3.13] The bound used to exclude |Z(G)| = 5 is incorrect. Corollary 2.2 for k = 5 gives |C_G(x)| ≤ 5 + |Z(G)|, so for |Z(G)| = 5 the allowed upper bound is 10, not 9 as stated in Lemma 3.13. Since |Z(G)| divides |C_G(x)| and |C_G(x)| > |Z(G)|, the value |C_G(x)| = 10 is possible. The class equation then admits solutions; for example n = 50 with nine classes of size 5 is arithmetically consistent. The stated argument therefore does not eliminate the |Z(G)| = 5 case, and a separate argument is required.
  4. [Corollaries 3.14 and 3.15] These corollaries are essential to Theorem 1.1 but are proved only by “similar computations” and no details are shown. For |Z(G)| = 2 with centralizer sizes 4 and 6, one must show that the class equation forces exactly the three groups D12, C4⋊C3, and SL(2,3). For |Z(G)| = 4 with all non-central centralizers of order 8, one must prove that exactly the six listed groups of order 16 arise and no others. These are nontrivial classification steps, especially since there are several groups of order 16. Lemma 3.10 similarly relies on an unstated classification of groups of order 18.
  5. [§4, Theorem 1.4] The proof of Theorem 1.4 contains the assertion that the set of all finite groups with a given number of conjugacy classes is finite, described as “clearly” true. This is a nontrivial theorem due to Landau and needs a citation or proof. Without this finiteness fact, the argument that each tuple (m1,...,mm) has finitely many preimages is incomplete. Since Theorem 1.4 is advertised in the abstract as one of the main results, this gap should be repaired.
minor comments (4)
  1. [Lemma 2.3] The proof of Lemma 2.3 concludes from t(m−k)=1 that m=n=1. The correct deduction is t=1 and hence n=m; one then still needs to say that |C_G(x)|=n for all x≠e forces every x to be central, contradicting the trivial-center assumption. The lemma is true, but the written proof is incomplete.
  2. [Lemma 3.3] In the display after “we have n−2/2 = k2n/5,” the equation should be n(5−2k2)=10, not n(5−nk2)=10 as printed. As written, the displayed equation is arithmetically inconsistent with the following conclusion k2=2 and n=10.
  3. [Theorem 1.1 and Corollary 1.2] The group (C4×C2)⋊C2 appears twice in the list in Theorem 1.1 and once more in the list in Corollary 1.2. The theorem should state a set of distinct groups or remove the duplicate.
  4. [Throughout] The phrase “strong k−star free” is typeset with a minus sign where a hyphen is intended, and the notation “5−star” is used inconsistently; a uniform notation such as “strong k-star free” would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classification follows from centralizer bounds and class-equation case analysis, not from its own conclusion.

full rationale

The paper's central derivation is not circular. Corollary 2.2 is a direct equivalence between strong k-star freeness and a uniform bound on non-central centralizer sizes, and the authors then enumerate all possible centralizer-size patterns for trivial center and use the class equation to restrict the group order. The candidate groups S3, D10, A4, GA(1,5), A5, D8, Q8, etc., are identified from the resulting orders and centralizer data using standard small-order classification or known structural facts; those facts are external inputs, not restatements of Theorem 1.1. There are no fitted parameters, no quantity is renamed as a prediction, and no load-bearing claim is justified by a self-citation: the references to previous commuting-graph work are contextual and do not supply the exclusion arguments. Some proof steps are abbreviated or potentially incomplete—Lemma 3.1 says 'The remaining cases are similar' for twenty exclusions, and Lemma 3.13's inequality uses |C_G(x)| <= 9 rather than the sharp |C_G(x)| <= 10 from Corollary 2.2—but those are correctness or completeness risks, not circularity. The argument would still be a genuine derivation even if some cases were checked more fully or found to be wrong.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities or parameters. It rests on standard classification facts and on an unproved finiteness assertion (Landau). The main mathematical input from the paper is the case analysis of the class equation, which is only partially shown.

assumptions (3)
  • standard math Classification of groups of order 18: there is no non-abelian group of order 18 with center order 3 and all non-central centralizers of order 6.
    Used in Lemma 3.10 to exclude n=18; no proof or citation is given.
  • standard math Finiteness of finite groups with a fixed number of conjugacy classes (Landau's theorem).
    Invoked in Theorem 1.4, stated as 'clearly this set is finite' without proof or reference.
  • standard math Classification of groups of order 16 used to enumerate groups with |Z(G)|=4 and |CG(x)|=8.
    Used in Lemma 3.15 and Corollary 3.15; the six listed groups are asserted without derivation.

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Pith. "Pith review of Groups with maximum vertex degree commuting graphs." pith.science (2026). https://pith.science/paper/MDFBPY4S

@misc{pith2026190808226,
  author       = {Pith},
  title        = {Pith review of: Groups with maximum vertex degree commuting graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDFBPY4S}},
  note         = {Machine review of arXiv:1908.08226}
}
abstract

Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph ${\Gamma}(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices are adjacent if they commute. We say that ${\Gamma}(G)$ is strong $k$ star free if the $k$ star graph is not a subgraph of ${\Gamma}(G)$. In this paper, we characterize all strong $5$ star free commuting graphs. As a byproduct, we classify all strong claw-free graphs. Also, we prove that the set of all non-abelian groups whose commuting graph is strong $k$ star free is finite.

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Reference graph

Works this paper leans on

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