arxiv: 2605.08607 · v1 · submitted 2026-05-09 · 🧮 math.GR

Recognition: no theorem link

On finite groups containing an element whose Engel sink is small

Evgeny Khukhro, Pavel Shumyatsky

Pith reviewed 2026-05-12 01:13 UTC · model grok-4.3

classification 🧮 math.GR

keywords finite groupsEngel sinksright Engel sinkleft Engel sinkcommutatorsgroup order boundscommutator generation

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The pith

Finite groups generated by commutators with one element have their order bounded by the size of that element's Engel sinks.

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

The paper shows that if a finite group G contains an element g such that G equals the subgroup generated by all commutators of g with elements of G, then the order of G is limited by the size of either a right or left Engel sink of g. A right Engel sink is a set that includes all sufficiently long repeated commutators starting from g, while a left sink does the same starting from other elements. This bound on group size follows from local information about how commutators stabilize. A sympathetic reader would care because it connects the generation of the whole group by commutators to a concrete numerical control on its cardinality.

Core claim

If a finite group G has an element g such that G=[G,g], then the order of G is bounded in terms of a right Engel sink of g, as well as in terms of a left Engel sink of g. The right sink contains all long enough commutators of the form [...[[g,x],x],...,x], and the left sink contains the corresponding left-normed versions [...[[x,g],g],...,g].

What carries the argument

The Engel sink of g, a subset containing all sufficiently long iterated commutators with g that measures the stabilization of repeated commutators under the condition that G equals its commutator subgroup with g.

If this is right

There is a function of the sink size that upper-bounds the order of G.
The same bounding function exists when using a left Engel sink instead of a right one.
The bound applies to every finite group meeting the commutator generation condition, without further restrictions on the group's structure.
Groups satisfying the hypothesis with a fixed sink size k must have order at most the value of that bounding function.

Where Pith is reading between the lines

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

For small fixed sink sizes one could in principle enumerate all possible groups up to the implied order bound.
The result suggests that Engel sinks of bounded size restrict the possible composition factors of G.
Analogous bounds might hold for other iterated commutator conditions in finite groups.

Load-bearing premise

That the complete list of finite simple groups is known and can be used to reduce the general case to a simpler one.

What would settle it

A finite group G with element g satisfying G=[G,g] where a right Engel sink of g has size at most k but the order of G exceeds any number determined solely by k.

read the original abstract

For an element $g$ of a group $G$, a right Engel sink of $g$ is a subset of $G$ containing all sufficiently long commutators $[...[[g ,x],x],\dots ,x]$ for all $x\in G$. A left Engel sink of $g$ is a subset of $G$ containing all sufficiently long commutators $[...[[x ,g ],g ],\dots ,g]$ for all $x\in G$. Using the classification of finite simple groups we prove that if a finite group $G$ has an element $g$ such that $G=[G,g]$, then the order of $G$ is bounded in terms of a right Engel sink of $g$, as well as in terms of a left Engel sink of $g$. Earlier Guralnick and Tracey proved this in the case where $g$ is an involution without using the classification.

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.

Desk Editor's Note private letter to a colleague

This extends the Guralnick-Tracey bound on group order from involutions to arbitrary elements g with G=[G,g] by using CFSG to reduce to the involution case, but the key check is whether the Engel sink size stays controlled through that reduction.

read the letter

The main result here is that if a finite group G has an element g with G equal to the commutator subgroup [G,g], then the order of G is bounded by a function of the size of a right Engel sink of g, and likewise for a left Engel sink. This builds on the earlier Guralnick-Tracey theorem that handled the special case where g is an involution and avoided the classification of finite simple groups entirely. The new work invokes CFSG to handle general g and produces the same style of bound. That extension is the actual advance, and the statement in the abstract is stated cleanly without extra claims. The paper does a straightforward job of citing the prior involution result and positioning the generalization inside the existing Engel-sink literature. No new technology or broad reshaping of group theory appears, which matches the modest scope. The potential soft spot is exactly the reduction step flagged in the stress-test note. The argument must produce an involution h satisfying G=[G,h] whose Engel sink size is bounded in terms of the original sink size for g; otherwise the claimed bound in terms of the given sink does not carry over. The abstract supplies no explicit reduction lemma or sink-size estimate, so the full manuscript needs to show that the quantitative control survives the passage to the involution case. If that link is made explicit and correct, the result stands; if it is only existential, the bound may be weaker than stated. This paper is aimed at specialists in finite group theory who already work with Engel conditions or commutator-generated subgroups. A reader familiar with the Guralnick-Tracey paper will see the incremental step right away and can judge the reduction details for themselves. It is worth sending to referees because the result is new within its narrow area and the approach follows standard methods for finite groups, even though it relies on CFSG. I would recommend peer review rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that if a finite group G contains an element g with G=[G,g], then |G| is bounded by a function of the size of any right Engel sink of g (and likewise for any left Engel sink). The argument proceeds by applying the Classification of Finite Simple Groups to reduce the general case to the involution case, which was already settled by Guralnick and Tracey without CFSG.

Significance. The result supplies a quantitative bound on group order in terms of Engel-sink cardinality under the commutator condition G=[G,g]. This extends the involution theorem in a natural direction and adds to the body of work on Engel conditions and sinks in finite groups. The use of CFSG is standard for such structural questions, but the value of the theorem rests on whether the reduction step preserves control over sink size.

major comments (2)

[§3] §3 (Reduction to the involution case): the argument invokes CFSG to produce an involution h with G=[G,h], yet provides no explicit estimate showing that the size of a right (or left) Engel sink of h is bounded by a function of the corresponding sink size for the original element g. Without such a quantitative transfer, the bound claimed for general g does not follow from the Guralnick–Tracey result.
[§4] §4 (Application to simple groups): the handling of non-abelian simple groups under the assumption G=[G,g] relies on the classification but does not verify that the Engel-sink cardinality remains controlled after the reduction; this step is load-bearing for the main theorem.

minor comments (2)

[Introduction] The introduction would benefit from a brief statement of the precise functional dependence (even if only existential) that the bound is claimed to have on the sink size.
[Theorem 1.1] Notation for left and right Engel sinks is introduced clearly but could be repeated once in the statement of the main theorem for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the insightful comments. We address the major comments below and plan to revise the paper to incorporate the necessary clarifications.

read point-by-point responses

Referee: [§3] §3 (Reduction to the involution case): the argument invokes CFSG to produce an involution h with G=[G,h], yet provides no explicit estimate showing that the size of a right (or left) Engel sink of h is bounded by a function of the corresponding sink size for the original element g. Without such a quantitative transfer, the bound claimed for general g does not follow from the Guralnick–Tracey result.

Authors: We agree with the referee that an explicit quantitative transfer between the Engel sink sizes of g and h is required to deduce the bound for general elements from the involution case. In the revised manuscript, we will insert a new subsection or lemma in §3 that establishes such a bound. The construction of h via the Classification will be analyzed to show that the right Engel sink of h has size bounded by a function of the right Engel sink of g (and analogously for left sinks), using the fact that long commutators with h can be related to those with g through the commutator condition G=[G,g]. This will complete the argument. revision: yes
Referee: [§4] §4 (Application to simple groups): the handling of non-abelian simple groups under the assumption G=[G,g] relies on the classification but does not verify that the Engel-sink cardinality remains controlled after the reduction; this step is load-bearing for the main theorem.

Authors: We thank the referee for this observation. The application in §4 to simple groups does rely on the classification to reduce to cases where the involution result applies, but we did not explicitly track the sink sizes in the write-up. We will revise §4 to include a verification that the Engel sink cardinalities for the reduced involutions are controlled by a function of the original sink size for g, drawing on the known structure of non-abelian simple groups satisfying G=[G,g]. This will strengthen the proof and address the concern. revision: yes

Circularity Check

0 steps flagged

No circularity: external CFSG reduction to independent prior theorem

full rationale

The derivation applies the Classification of Finite Simple Groups (an external, independently established theorem) to reduce the general case to the involution case, which is handled by a theorem of Guralnick and Tracey (distinct authors). The claimed bound is existential (order bounded by some function of sink size) rather than a fitted parameter or self-referential definition. No self-citations are load-bearing, no ansatz is smuggled, and no step reduces the result to its own inputs by construction. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof invokes the classification of finite simple groups as an external theorem and reduces the general case to the involution case already proved by Guralnick and Tracey. No free parameters or new entities are introduced in the abstract statement.

axioms (1)

standard math Classification of Finite Simple Groups
Invoked to handle non-involution elements after reducing to the known involution case.

pith-pipeline@v0.9.0 · 5456 in / 1098 out tokens · 32971 ms · 2026-05-12T01:13:32.936585+00:00 · methodology

discussion (0)

Reference graph

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