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arxiv: 2605.14862 · v1 · pith:JWQCDPOZnew · submitted 2026-05-14 · 🧮 math.GR

Commutative decomposition of infinite symmetric groups and transformation monoids

Luna Elliott , Alex Levine This is my paper

Pith reviewed 2026-06-30 19:44 UTC · model grok-4.3

classification 🧮 math.GR
keywords symmetric grouptransformation monoidcommutative widthabelian subgroupsinfinite setsgroup decompositionmonoid decomposition
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The pith

Sym(N) equals a product of at most nine abelian subgroups.

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

The paper determines the commutative subgroup width of the symmetric group on the natural numbers and the corresponding widths for several transformation monoids. It proves that Sym(N) is the product of nine abelian subgroups, improving the prior upper bound of fourteen, and shows that four is a lower bound. For the full transformation monoid, the partial transformation monoid, and the symmetric inverse monoid on N, the commutative submonoid width is exactly three. It further shows that the commutative inverse submonoid width is infinite for every infinite symmetric inverse monoid. The constructions use the countability of the underlying set to produce the required factors explicitly.

Core claim

The commutative subgroup width of Sym(N) is at most 9. The commutative submonoid widths of the full transformation monoid N^N, the partial transformation monoid P_N and the symmetric inverse monoid I_N are exactly 3. The commutative inverse submonoid width of any infinite symmetric inverse monoid is always infinite.

What carries the argument

Commutative subgroup width: the smallest k such that the group equals a product of k abelian subgroups.

If this is right

  • Sym(N) decomposes into a product of nine abelian subgroups.
  • N^N, P_N and I_N each decompose into a product of three commutative submonoids.
  • Every infinite symmetric inverse monoid requires infinitely many commutative inverse submonoids in any such decomposition.
  • The widths are finite precisely when the set is countable.

Where Pith is reading between the lines

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

  • For uncountable base sets the same finite widths may fail because the countable-support constructions no longer apply.
  • The gap between the lower bound of 4 and upper bound of 9 for Sym(N) leaves open the possibility that the exact width is strictly between those numbers.
  • The exact value 3 for the monoids suggests that the presence of non-bijective elements simplifies the commutative decomposition relative to the group case.

Load-bearing premise

The underlying set is countably infinite, which permits explicit constructions of the abelian or commutative factors via countable supports.

What would settle it

An explicit collection of eight abelian subgroups whose product fails to be all of Sym(N), or a demonstration that three commutative submonoids cannot multiply to N^N.

read the original abstract

The commutative subgroup width of a group $G$ is the smallest $k$ such that there are abelian subgroups $A_0,A_1,\ldots,A_{k-1}\leq G$ with $G=A_0A_1\cdots A_{k-1}$. Commutative (inverse) submonoid width is defined analogously. In 2002, Ab\'{e}rt showed, rather surprisingly, that the commutative subgroup width of the symmetric group on an infinite set is always finite. It was later shown by Seress that it is always bounded above by $14$. We answer a question of Seress and show that in fact the commutative subgroup width of $\operatorname{Sym}(\mathbb{N})$ is at most $9$. We improve the best known lower bound to $4$. We also study standard monoid analogues of the symmetric group; showing that the commutative submonoid widths of the full transformation monoid $\mathbb{N}^\mathbb{N}$, the partial transformation monoid $P_\mathbb{N}$ and the symmetric inverse monoid $I_\mathbb{N}$ are exactly $3$. We conclude by showing that the commutative inverse submonoid width of any infinite symmetric inverse monoid is always infinite.

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 / 4 minor

Summary. The manuscript defines the commutative subgroup width of a group G as the smallest k such that G is a product of k abelian subgroups, and analogously for commutative (inverse) submonoid width. It proves that the commutative subgroup width of Sym(ℕ) is at most 9 (improving Seress's bound of 14) and at least 4, answering a question of Seress. It further shows that the commutative submonoid widths of the full transformation monoid ℕ^ℕ, the partial transformation monoid P_ℕ, and the symmetric inverse monoid I_ℕ are exactly 3, and that the commutative inverse submonoid width of any infinite symmetric inverse monoid is infinite.

Significance. If the explicit constructions and lower-bound arguments hold, the results refine the understanding of abelian and commutative factorizations in infinite symmetric groups and transformation monoids. The improvement to an upper bound of 9, the exact determination of width 3 for the three monoids, and the infinitude result for inverse submonoids are concrete advances; the countable-support constructions for the upper bounds on ℕ are a particular strength.

minor comments (4)
  1. §1: The citation to Abért (2002) and Seress should include the precise theorem or question number being answered for easier cross-reference.
  2. §3, Definition 3.2: The notation for partial transformations in P_ℕ could be aligned more closely with standard references (e.g., Howie) to avoid minor ambiguity in the support condition.
  3. §5: The infinitude proof for inverse submonoids relies on an infinite descending chain argument; a brief remark on why the same argument fails for the non-inverse case would clarify the distinction.
  4. Throughout: A small number of typographical inconsistencies appear in the use of blackboard-bold versus script letters for the monoids; these do not affect readability but should be standardized.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation of minor revision. The referee's description of the results is accurate.

Circularity Check

0 steps flagged

No significant circularity; explicit constructions and proofs are self-contained

full rationale

The paper's central results consist of explicit upper-bound constructions for the commutative widths of Sym(N), N^N, P_N and I_N on the countable set N, together with lower-bound arguments and an infinitude proof for inverse submonoids. These are presented as direct consequences of countable-support factorizations and combinatorial arguments rather than any fitted parameters, self-definitions, or load-bearing self-citations. No equation or definition reduces a claimed width to a quantity defined in terms of itself, and the cited prior results (Abért, Seress) are external. The derivation chain therefore remains independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely within the standard axioms of ZFC set theory and the definitions of groups and monoids; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms of ZFC set theory and the definition of the symmetric group and transformation monoids on an infinite set
    Invoked to guarantee the existence of the underlying infinite set N and the algebraic structures Sym(N), N^N, etc.

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Reference graph

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