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arxiv: 2408.03879 · v5 · submitted 2024-08-07 · 🧮 math.GR · math.CO

Engel and co-Engel graphs of finite groups

Peter J. Cameron , Rishabh Chakraborty , Rajat Kanti Nath , Deiborlang Nongsiang This is my paper

Pith reviewed 2026-05-23 22:08 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords co-Engel graphEngel graphfinite groupsFitting subgrouptoroidal graphprojective graphclique numbergraph invariants
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The pith

Finite non-Engel groups whose co-Engel subgraphs on non-Fitting elements have clique number at most 4 are classified when toroidal or projective.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper associates to each group an Engel digraph whose arcs record the existence of an iterated commutator equal to the identity and defines the co-Engel graph as the complement of the underlying undirected graph. It establishes that the isolated vertices of the co-Engel graph are precisely the elements of the Fitting subgroup. For selected finite non-Engel groups the induced subgraph on the complement of the Fitting subgroup is constructed explicitly and its genus, spectra, energies and Zagreb indices are computed. These calculations produce a complete list, up to isomorphism, of all finite non-Engel groups for which the induced subgraph has clique number at most 4 and is toroidal or projective. The same subgraphs are shown to be ALQ-integral and to satisfy the E-LE and Hansen-Vukičević conjectures.

Core claim

We realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups G induced by G minus F(G). We also compute genus, various spectra, energies and Zagreb indices of E_c^-(G) for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group G such that the clique number of E_c^-(G) is at most 4 and E_c^-(G) is toroidal or projective. Further, we show that E_c^-(G) is ALQ-integral and satisfies the E-LE conjecture and the Hansen-Vukičević conjecture for the groups considered in this paper.

What carries the argument

The induced co-Engel subgraph E_c^-(G) on G minus the Fitting subgroup F(G), whose edges record pairs of elements that fail to satisfy any iterated commutator identity.

If this is right

  • All finite non-Engel groups satisfying the clique-number and surface conditions on E_c^-(G) are determined up to isomorphism.
  • The graphs E_c^-(G) admit explicit computations of genus, spectra, energies and Zagreb indices.
  • E_c^-(G) is ALQ-integral for every group considered.
  • E_c^-(G) satisfies both the E-LE conjecture and the Hansen-Vukičević conjecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The universality result for Engel digraphs shows that arbitrary finite directed graphs can appear as induced subdigraphs inside the commutator structure of some finite group.
  • The scarcity of groups in which the undirected Engel graph fails to recover the directed version indicates that directed information is typically recoverable from the undirected graph for most groups of small order.
  • The restriction to toroidal and projective surfaces supplies a topological filter that may extend to other surfaces once the classification method is generalized.

Load-bearing premise

The isolated vertices of the co-Engel graph E_c(G) are exactly the elements of the Fitting subgroup F(G).

What would settle it

A finite non-Engel group G outside the listed isomorphism classes whose induced subgraph E_c^-(G) has clique number at most 4 and is toroidal or projective.

Figures

Figures reproduced from arXiv: 2408.03879 by Deiborlang Nongsiang, Peter J. Cameron, Rajat Kanti Nath, Rishabh Chakraborty.

Figure 1
Figure 1. Figure 1: Embedding of E −(C3 × D6) on a torus. ① ① ① ① ①① ① ① PPPPPPPPPPPPPPPPPPPPP ❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜ ✚ ✚✚✚ ✚✚✚✚ ✚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❩ ❩❩❩ ❩❩❩❩ ❩ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧✧ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✥✥✥ ✥✥✥ ✥✥✥ ✥✥✥ ✥✥✥ ✥✥✥ ✥✥✥ ✥✥✥ ✥✥✥ ✥✥✥ ✑✑✑✑✑✑✑✑ ✑✑ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏✏ ✏ ❵❵❵ ❵❵❵ ❵❵❵ ❵❵❵ ❵❵❵ ❵❵❵ ❵❵❵ ❵❵❵ ❵❵❵ ❵❵❵ ❜ ❜… view at source ↗
Figure 2
Figure 2. Figure 2: E −(A4) Let z ∈ Z ∗ (G), z 6= 1. Then, by Lemma 4.12, every elements of H¯ ∪ zH¯ is adjacent to every elements of K¯ ∪ zK¯ , showing that K8,8 is a subgraph of E −(G) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Embedding of E −(A4) on a torus. Thus γ(E −(G)) ≥ γ(K8,8) = 9, a contradiction. Thus |Z ∗ (G)| = 1 and so G ∼= A4. An embedding of the graph E −(A4) on a torus is shown in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

Let $G$ be a group. Associate a directed graph $\vec{E}(G)$ (called the Engel digraph of $G$) with $G$ whose vertex set is $G$, with an arc $(x,y)$ if $[y, {}_k x]=1$ for some positive integer $k$, where $[y,{}_kx]$ is the iterated commutator $[y,x,x,\ldots,x]$, with $k$ terms $x$ in the expression. From this we define the Engel graph $E(G)$ by ignoring directions; the co-Engel graph $E_c(G)$ is its complement. The co-Engel graph, under the name ``Engel graph'', was introduced by Abdollahi. However, the name we use is more natural. We begin with some general results about the Engel digraph and graph, before turning our attention to the co-Engel graph. Among other things, we show that the undirected Engel graph does not determine the directed version up to isomorphism, though counterexamples seem to be fairly rare: there are just two orders less than $100$ for which this happens. We also prove a universality theorem: every finite digraph is an induced sub-digraph of the Engel digraph of a finite group. The isolated vertices of $E_c(G)$ form the Fitting subgroup $F(G)$ of $G$. In this paper, we realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups $G$ induced by $G \setminus F(G)$. We write $E_c^-(G)$ to denote the subgraph of $E_c(G)$ induced by $G \setminus F(G)$. We also compute genus, various spectra, energies and Zagreb indices of $E_c^-(G)$ for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number of $E_c^-(G)$ is at most $4$ and $E_c^-(G)$ is toroidal or projective. Further, we show that $E_c^-(G)$ is ALQ-integral and satisfies the E-LE conjecture and the Hansen-Vuki{\v{c}}evi{\'c} conjecture for the groups considered in this paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper defines the Engel digraph and co-Engel graph E_c(G) for finite groups G. It establishes general properties of these graphs, including that the isolated vertices of E_c(G) coincide with the Fitting subgroup F(G). It then examines the induced subgraph E_c^-(G) on G minus F(G) for non-Engel groups, computes the genus, spectra, energies, and Zagreb indices of these graphs for specific families, and classifies (up to isomorphism) all finite non-Engel groups G such that the clique number of E_c^-(G) is at most 4 and E_c^-(G) is toroidal or projective. It further verifies that the graphs under consideration are ALQ-integral and satisfy the E-LE and Hansen-Vukičević conjectures.

Significance. If the central claims hold, the classification supplies a complete, explicit list of qualifying groups, which constitutes a concrete contribution to the study of Engel-type graphs on groups. The universality result that every finite digraph embeds as an induced subdigraph of some Engel digraph is a strong general theorem. The paper supplies explicit realizations together with computed invariants and conjecture verifications, which are positive features of the work.

major comments (2)
  1. [Abstract (the paragraph beginning 'The isolated vertices of E_c(G) form the Fitting subgroup F(G) of G')] The statement that the isolated vertices of E_c(G) are precisely F(G) is load-bearing for the definition of E_c^-(G) and for the subsequent classification of all qualifying non-Engel groups. The manuscript must contain a self-contained proof of this general fact (with explicit verification that no elements of F(G) fail to be isolated and that no elements outside F(G) are isolated), because any counterexample would render both the induced-subgraph constructions and the exhaustive list incomplete.
  2. [Abstract (the sentence beginning 'As a consequence, we determine (up to isomorphism) all finite non-Engel group G')] The classification result (clique number ≤4 and toroidal/projective) is derived from the groups whose E_c^-(G) are realized in the paper; without an explicit enumeration of those groups together with the corresponding graph computations, it is impossible to confirm that the list is exhaustive and that the genus/spectral claims hold for each.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below and will revise the paper accordingly to strengthen the presentation and verifiability of the results.

read point-by-point responses

  1. Referee: [Abstract (the paragraph beginning 'The isolated vertices of E_c(G) form the Fitting subgroup F(G) of G')] The statement that the isolated vertices of E_c(G) are precisely F(G) is load-bearing for the definition of E_c^-(G) and for the subsequent classification of all qualifying non-Engel groups. The manuscript must contain a self-contained proof of this general fact (with explicit verification that no elements of F(G) fail to be isolated and that no elements outside F(G) are isolated), because any counterexample would render both the induced-subgraph constructions and the exhaustive list incomplete.

    Authors: We agree that a fully self-contained proof of this property is necessary for the subsequent constructions. While the manuscript states the result, we will add a dedicated lemma with a complete, self-contained proof in the revised version. The proof will explicitly verify both directions: every element of F(G) is isolated in E_c(G), and no element outside F(G) is isolated. This addresses the concern directly and ensures the definition of E_c^-(G) rests on solid ground. revision: yes

  2. Referee: [Abstract (the sentence beginning 'As a consequence, we determine (up to isomorphism) all finite non-Engel group G')] The classification result (clique number ≤4 and toroidal/projective) is derived from the groups whose E_c^-(G) are realized in the paper; without an explicit enumeration of those groups together with the corresponding graph computations, it is impossible to confirm that the list is exhaustive and that the genus/spectral claims hold for each.

    Authors: We acknowledge that an explicit enumeration would improve verifiability of the classification and the associated computations. In the revised manuscript we will include a clear table or list enumerating all qualifying groups up to isomorphism, together with the specific computations of E_c^-(G) (clique number, genus, spectra, etc.) for each. This will make the exhaustiveness of the list and the supporting calculations transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on standard definitions and independent proofs

full rationale

The paper defines Engel and co-Engel graphs via iterated commutators and their complements using standard group operations. The statement that isolated vertices of E_c(G) are exactly F(G) is asserted as a general fact before defining E_c^-(G); this is a load-bearing definitional step but is not shown to reduce to itself by construction, nor does it rely on self-citation chains, fitted parameters, or ansatzes smuggled from prior work. No equations, predictions, or uniqueness theorems appear that collapse the central classification result to its inputs. The derivation chain is self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters or new postulated entities; it works entirely within the standard axioms of finite group theory and undirected graph theory.

axioms (1)
  • standard math Finite groups obey the standard axioms of associativity, identity element, and inverses, allowing definition of commutators and the Fitting subgroup.
    All graph constructions and the identification of isolated vertices rely on these background facts.

pith-pipeline@v0.9.0 · 5981 in / 1361 out tokens · 47071 ms · 2026-05-23T22:08:11.948366+00:00 · methodology

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Reference graph

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