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arxiv: 2605.22302 · v1 · pith:ROX7DJTUnew · submitted 2026-05-21 · 🧮 math.GR

On finite perfect two-sided skew braces

Marco Damele This is my paper

Pith reviewed 2026-05-22 02:17 UTC · model grok-4.3

classification 🧮 math.GR
keywords skew bracestwo-sided skew bracesperfect groupscentral productsfinite algebraic structuresquasi-simple bracesgroup decompositions
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The pith

Every finite perfect two-sided skew brace decomposes as a central product of an almost trivial skew brace with perfect additive group and a trivial skew brace with perfect additive group.

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

The paper establishes a structure theorem for finite perfect two-sided skew braces by introducing a central product theory that works in equivalent external and internal forms. This leads to a canonical decomposition in which one factor is almost trivial and the other is trivial, both built from perfect groups. The result yields a classification of such braces up to central amalgamation. It further shows that perfectness of the brace is equivalent to perfectness of its additive group or its multiplicative group. The decomposition recovers prior classifications when the center is trivial and proves that quasi-simple two-sided skew braces must be trivial or almost trivial.

Core claim

Every finite perfect two-sided skew brace B admits the canonical decomposition B = B² ◦ B^{2,op}, where B² is almost trivial with perfect additive group, while B^{2,op} is trivial with perfect additive group. Thus finite perfect two-sided skew braces are classified, up to central amalgamation, by trivial and almost trivial skew braces arising from perfect groups. For finite two-sided skew braces, perfectness of the skew brace is equivalent to perfectness of either the additive or the multiplicative group.

What carries the argument

The central product construction for skew braces, developed in both external and internal forms and shown to be equivalent, which enables the decomposition of finite perfect two-sided cases.

If this is right

  • Finite perfect two-sided skew braces are classified up to central amalgamation by trivial and almost trivial skew braces arising from perfect groups.
  • For finite two-sided skew braces, perfectness of the brace is equivalent to perfectness of either the additive group or the multiplicative group.
  • When the center is trivial the central product reduces to a direct product, recovering the classification of finite simple two-sided skew braces.
  • Quasi-simple two-sided skew braces must be either trivial or almost trivial.

Where Pith is reading between the lines

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

  • The two-sided condition appears essential for the rigidity that forces quasi-simple examples to be trivial or almost trivial.
  • The construction of a quasi-simple skew brace that is not two-sided suggests that without two-sidedness the decomposition may fail.
  • Similar central-product arguments could be tested on infinite skew braces or on other related algebraic structures if the theory extends.

Load-bearing premise

The central product theory for skew braces applies validly to finite perfect two-sided cases and produces the stated decomposition.

What would settle it

A finite perfect two-sided skew brace that cannot be expressed as such a central product of an almost trivial part and a trivial part, both with perfect additive groups, would disprove the main theorem.

read the original abstract

We prove a structure theorem for finite perfect two-sided skew braces. The main tool is a central product theory for skew braces, developed here in both external and internal form; we show that these two constructions are equivalent. Our main result states that every finite perfect two-sided skew brace \(B\) admits the canonical decomposition $B=B^2\circ B^{2,\operatorname{op}},$ where \(B^2\) is almost trivial with perfect additive group, while \(B^{2,\operatorname{op}}\) is trivial with perfect additive group. Thus finite perfect two-sided skew braces are classified, up to central amalgamation, by trivial and almost trivial skew braces arising from perfect groups. This decomposition has strong consequences for the underlying groups: for finite two-sided skew braces, perfectness of the skew brace is equivalent to perfectness of either the additive or the multiplicative group. In the trivial-center case the central product becomes a direct product, recovering Trappeniers' classification of finite simple two-sided skew braces. We also show that quasi-simple two-sided skew braces are necessarily either trivial or almost trivial. Finally, we prove that this rigidity is genuinely two-sided by constructing a quasi-simple skew brace which is not two-sided and is neither trivial nor almost trivial.

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 presents a structure theorem for finite perfect two-sided skew braces. The authors develop a central product theory for skew braces in both external and internal forms and prove their equivalence. The central result states that every finite perfect two-sided skew brace B admits the canonical decomposition B = B² ◦ B^{2,op}, where B² is almost trivial with perfect additive group and B^{2,op} is trivial with perfect additive group. This yields a classification of such braces up to central amalgamation by trivial and almost trivial skew braces arising from perfect groups. Additional results include the equivalence, for finite two-sided skew braces, of skew-brace perfectness with perfectness of either the additive or multiplicative group; recovery of Trappeniers' classification of finite simple two-sided skew braces in the trivial-center case; the fact that quasi-simple two-sided skew braces are necessarily trivial or almost trivial; and an explicit construction of a quasi-simple skew brace that is not two-sided and is neither trivial nor almost trivial.

Significance. If the central-product construction and the ensuing decomposition are valid, the paper supplies a substantial classification result for finite perfect two-sided skew braces and introduces a new technical tool (central products for skew braces) that may be useful in wider investigations of skew-brace structure. The equivalence between brace-level and group-level perfectness, the recovery of the earlier simple-case classification, and the rigidity statement for quasi-simple objects all sharpen the distinction between two-sided and general skew braces. The counter-example for the non-two-sided case usefully illustrates the necessity of the two-sided hypothesis.

minor comments (4)
  1. Abstract: the phrase 'almost trivial' is introduced without a parenthetical gloss or forward reference to its definition; a single sentence clarifying the notion would improve immediate readability for readers outside the immediate subfield.
  2. Section 3 (central-product theory): the equivalence proof between external and internal constructions is stated to hold under the finite hypothesis; a brief remark on whether any step genuinely requires finiteness (or whether the equivalence extends verbatim to the infinite case) would strengthen the exposition.
  3. Section 5 (consequences for perfectness): the claim that perfectness of the brace is equivalent to perfectness of either underlying group is central; an explicit small-order example illustrating the failure of the equivalence when the two-sided condition is dropped would make the statement more concrete.
  4. References: the citation to Trappeniers' classification should include the precise journal, volume, and year to facilitate verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. We appreciate the recognition given to the central-product construction, the structure theorem, and the consequences for perfectness and quasi-simplicity in the two-sided setting.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper first develops a central product theory for skew braces in both external and internal forms, proves their equivalence as a new construction, and only then applies it to obtain the canonical decomposition B = B² ◦ B^{2,op} for finite perfect two-sided cases. All subsequent claims (equivalence of brace perfectness to group perfectness, classification up to central amalgamation, and rigidity results) follow directly from this independent tool under the finite two-sided hypothesis. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to the target result itself; the central product is introduced and validated prior to its use on perfect braces.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background from group theory and skew brace definitions; no free parameters are introduced, no new entities are postulated, and the central product construction is developed within the paper rather than assumed ad hoc.

axioms (1)
  • standard math Standard axioms and basic properties of groups, skew braces, perfect groups, and normal subgroups
    Invoked throughout the development of central products and the decomposition theorem.

pith-pipeline@v0.9.0 · 5739 in / 1446 out tokens · 61471 ms · 2026-05-22T02:17:47.334559+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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