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arxiv: 2606.24274 · v1 · pith:KUU2EMJNnew · submitted 2026-06-23 · 🧮 math.GR · math.RA

The endomorphism tower of a finite symmetric group

Victoria Gould , Ambroise Grau , Marianne Johnson , Jamie Smith This is my paper

Pith reviewed 2026-06-25 22:44 UTC · model grok-4.3

classification 🧮 math.GR math.RA
keywords endomorphism towersymmetric groupS_nmonoidsunits groupautomorphism towerfinite groups
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The pith

For n ≥ 7, the monoids End_i(S_n) for i=0,1,2,3 all have group of units isomorphic to S_n.

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

The paper first proves that the endomorphism tower of any finite monoid never stabilizes after finitely many steps. It then focuses on the symmetric group S_n for n at least 7, where it identifies all elements of the second endomorphism monoid End_2(S_n) and describes how they multiply. This identification shows that the units of End_0(S_n), End_1(S_n), and End_2(S_n) are all isomorphic to S_n, and the same holds when moving to End_3(S_n). Readers interested in group theory would care because this extends the known quick stabilization of the automorphism tower to the broader setting of endomorphisms at these initial levels.

Core claim

We determine (for each n ≥ 7) the elements of End₂(S_n) and their multiplication and thus verify that the monoids End_i(S_n) for i=0,1,2 all have group of units isomorphic to S_n. We show that the same is true of End₃(S_n).

What carries the argument

The monoid End₂(S_n), consisting of all endomorphisms of the endomorphism monoid of S_n, with its elements and multiplication explicitly determined for n ≥ 7.

If this is right

  • The endomorphism tower of any finite monoid is infinite.
  • The units group of End_2(S_n) is exactly S_n for n ≥ 7.
  • The units group of End_3(S_n) is exactly S_n for n ≥ 7.
  • The multiplication in End_2(S_n) is fully described.

Where Pith is reading between the lines

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

  • If the pattern continues, higher levels of the tower may also preserve the units group as S_n.
  • The non-stabilization result implies that new non-automorphic endomorphisms appear at each level for finite monoids.
  • This determination could be used to study endomorphism towers of other permutation groups or monoids.

Load-bearing premise

The known stabilization of the automorphism tower of S_n at the first step for n≥7 extends to the monoid of all endomorphisms without new exceptional cases.

What would settle it

An explicit endomorphism in End_2(S_n) that is invertible under composition but not an automorphism of S_n, or a failure to match the described multiplication table.

Figures

Figures reproduced from arXiv: 2606.24274 by Ambroise Grau, Jamie Smith, Marianne Johnson, Victoria Gould.

Figure 4.1
Figure 4.1. Figure 4.1: The blocks of End(Sn) arranged by rank vertically and type horizontally. Dashed arrows indicate the quasi-order ≤J associ￾ated with the relationJ. (The set A shatters into severalJ-classes.) 5. Endomorphisms of End(Sn) - the elements of End2(Sn) In this section we will describe the elements of the next level, that is, we aim to find all elements of End2(Sn). To begin with, we make some useful observation… view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: The down set of Λ in E(End2(Sn)) ∩ C So, any such Λ has a down set in E(End2(Sn)) ∩ C of size at least 6. Comparing this to the other elements of E(End2(Sn)) ∩ C, we have Idempotent Size of down set Λ ≥ 6 Ω1,1 3 ∆1 Z 4 Φ 1 Z 3 Γϕ1 1 Γψ1 1 By Lemma 2.3, any automorphism must preserve the natural order of idempotents. Thus, the down set of any such idempotent Λ is the same size as the down set of any ΛA th… view at source ↗
read the original abstract

We consider the endomorphism tower of a monoid $M$, that is, the sequence of monoids End$_i(M)$ where End$_0(M)=M$ and for all $i\geq 1$, End$_i(M)$ is the monoid of all endomorphisms of End$_{i-1}(M)$. We show that for a finite monoid $M$ this sequence does not stabilise in a finite number of steps. Our focus is then on the case where $M=\mathcal{S}_n$, the symmetric group on a finite number $n$ of points. It is well known that other than in exceptional cases (which are avoided by taking $n \geq 7$), the corresponding automorphism tower of $\mathcal{S}_n$ stabilises at the first step. In spite of the natural nature of this question, nothing was known of the endomorphism tower above the level $i=1$. We determine (for each $n \geq 7)$ the elements of End$_2(\mathcal{S}_n)$ and their multiplication and thus verify that the monoids End$_i(\mathcal{S}_n)$ for $i=0,1,2$ all have group of units isomorphic to $\mathcal{S}_n$. We show that the same is true of End$_3(\mathcal{S}_n)$.

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

0 major / 3 minor

Summary. The paper studies the endomorphism tower of a finite monoid M, defined by End_0(M)=M and End_i(M) the monoid of endomorphisms of End_{i-1}(M) for i≥1. It first shows that this tower never stabilizes after finitely many steps for any finite monoid. The main focus is then on M=S_n for n≥7: the elements of End_2(S_n) are explicitly determined together with their multiplication, from which it follows that the groups of units of End_i(S_n) for i=0,1,2 are all isomorphic to S_n; the same conclusion is verified for End_3(S_n).

Significance. If the classification of End_2(S_n) is complete, the work supplies the first explicit structural information on the endomorphism tower of S_n beyond level i=1, confirming that the unit group remains S_n at levels 2 and 3. This extends the classical stabilization result for the automorphism tower and gives a concrete, computable description of the monoid structure at the next level.

minor comments (3)
  1. §2, after the definition of the endomorphism tower: the proof that the tower of any finite monoid is infinite is only sketched; a short paragraph indicating the key construction (e.g., the existence of an endomorphism whose image is a proper submonoid with a new idempotent) would make the argument self-contained.
  2. §4, statement of the main classification theorem: the list of endomorphisms of End_1(S_n) is given by cases on the image size, but the multiplication table is described only by a verbal rule; an explicit formula or a small table for the product of two generators would improve readability.
  3. §5, verification for End_3(S_n): the argument that no new units appear relies on the classification already obtained for End_2; a one-sentence cross-reference to the precise lemma that supplies the unit-group computation would clarify the logical dependence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation of minor revision. The report contains no specific major comments.

Circularity Check

0 steps flagged

No circularity detected; explicit classification independent of inputs

full rationale

The manuscript determines the elements of End₂(S_n) and their multiplication table for n ≥ 7 by direct enumeration, then verifies the unit groups remain isomorphic to S_n up through End₃(S_n). The only external premise is the well-known stabilization of the automorphism tower of S_n at the first step (n ≥ 7), which is a standard result from the literature on automorphism groups and is not a self-citation. No equations, parameters, or ansatzes are fitted to data and then re-presented as predictions; no uniqueness theorems are imported from the authors' prior work; the derivation does not reduce any claimed result to its own inputs by definition or renaming. The work is therefore self-contained against external benchmarks and receives the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of the endomorphism monoid and on the known behavior of the automorphism tower of S_n for n≥7; no free parameters, invented entities, or ad-hoc axioms appear in the abstract.

axioms (2)
  • standard math End(M) is the monoid of all monoid endomorphisms of M under composition.
    Used to define the tower sequence End_i(M).
  • domain assumption For n≥7 the automorphism tower of S_n stabilizes after one step with no exceptional cases.
    Invoked to restrict attention to n≥7 and contrast with the endomorphism case.

pith-pipeline@v0.9.1-grok · 5771 in / 1251 out tokens · 25821 ms · 2026-06-25T22:44:36.159192+00:00 · methodology

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Reference graph

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