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arxiv: 1907.00508 · v1 · pith:EWXKI2UMnew · submitted 2019-07-01 · 🧮 math.GR

Finiteness conditions for the weak commutativity construction

Raimundo Bastos , Bruno Lima , Ricardo Nunes This is my paper

Pith reviewed 2026-05-25 11:57 UTC · model grok-4.3

classification 🧮 math.GR
keywords weak commutativitylocally finite groupsfinite exponentfiniteness criteriaD(G) subgroupcommutator constructionsgroup operators
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The pith

If G is locally finite with finite exponent n, then χ(G) is locally finite with finite exponent bounded in terms of n.

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

The paper proves that the weak commutativity operator χ preserves local finiteness and finite exponent when the input group satisfies those conditions. If G is locally finite of exponent n, then the generated group χ(G) is locally finite and its exponent is finite and controlled by n. This extends earlier results on preservation of finiteness, solubility, and nilpotency. The authors also give criteria for when the commutator subgroup D(G) inside χ(G) is finite, based on the set of the generating commutators.

Core claim

If G is a locally finite group with exp(G)=n, then χ(G) is locally finite and has finite n-bounded exponent. Further, we examine some finiteness criteria for the subgroup D(G) = ⟨[g1,g2^φ] | gi ∈ G⟩ ≤ χ(G) in terms of the set {[g1,g2^φ] | gi ∈ G}.

What carries the argument

The weak commutativity operator χ(G) = ⟨G ∪ G^φ | [g, g^φ]=1 for all g in G⟩, together with the derived subgroup D(G) generated by commutators between the two copies.

If this is right

  • χ(G) is locally finite whenever G is locally finite of finite exponent.
  • The exponent of χ(G) is finite and depends only on the exponent n of G.
  • Finiteness of D(G) follows from suitable conditions on the set of commutators [g1, g2^φ].
  • The construction extends known preservation results from finite groups to the locally finite case.

Where Pith is reading between the lines

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

  • The result supplies a systematic way to produce new locally finite groups of finite exponent from existing ones.
  • Similar transfer arguments might apply to other group-theoretic operators defined by adjoining isomorphic copies with commutation relations.
  • Explicit bounds on the exponent of χ(G) could be computed for concrete families such as finite p-groups.

Load-bearing premise

The input group G must itself be locally finite and have finite exponent n.

What would settle it

A locally finite group G of finite exponent n such that χ(G) is not locally finite or has exponent not bounded by any function of n.

read the original abstract

The operator, $\chi $, of weak commutativity between isomorphic groups $G$ and $G^{\varphi }$ was introduced by Sidki as \begin{equation*} \chi (G)=\left\langle G \cup G^{\varphi }\mid \lbrack g,g^{\varphi }]=1\,\forall \,g\in G\right\rangle \text{.} \end{equation*} It is known that the operator $\chi $ preserves group properties such as finiteness, solubility and also nilpotency for finitely generated groups. We prove that if $G$ is a locally finite group with $exp(G)=n$, then $\chi(G)$ is locally finite and has finite $n$-bounded exponent. Further, we examine some finiteness criteria for the subgroup $D(G) = \langle [g_1,g_2^{\varphi}] \mid g_i \in G\rangle \leqslant \chi(G)$ in terms of the set $\{[g_1,g_2^{\varphi}] \mid g_i \in G\}$.

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 / 2 minor

Summary. The manuscript studies the weak commutativity operator χ(G), defined by the presentation ⟨G ∪ G^φ | [g, g^φ]=1 ∀ g∈G⟩ where G^φ is an isomorphic copy of G. It proves that if G is locally finite of exponent n then χ(G) is locally finite of exponent bounded by a function of n. It further gives finiteness criteria for the subgroup D(G) generated by the set {[g1,g2^φ] | gi∈G} in terms of that set.

Significance. The result extends the known preservation properties of χ (finiteness, solubility, nilpotency for finitely generated groups) to the important class of locally finite groups of finite exponent. Because the construction is local—any finite set of generators of χ(G) lies in the image of a finitely generated subgroup of G—the transfer of local finiteness and bounded exponent follows from the corresponding properties in finite groups of exponent dividing n. The additional analysis of D(G) supplies concrete criteria that may be useful in further structural work on χ(G).

minor comments (2)
  1. [Abstract] Abstract, displayed equation: the notation exp(G)=n is used without prior definition; a parenthetical remark that exp denotes the exponent would aid readers unfamiliar with the abbreviation.
  2. The statement that χ preserves nilpotency only for finitely generated groups is mentioned in passing; a reference to the relevant earlier result would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The summary accurately captures the main results on the preservation of local finiteness and bounded exponent under the weak commutativity operator, as well as the criteria for D(G).

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes an implication: local finiteness and exponent n on G transfer to χ(G) via the explicit presentation χ(G) = <G ∪ G^φ | [g, g^φ]=1 ∀g∈G>. This is a standard preservation argument under the defining relations; no equation reduces the conclusion to a redefinition of the inputs, no parameter is fitted then renamed as a prediction, and the cited prior facts about χ preserving finiteness/solubility/nilpotency are external to the authors. The derivation chain is self-contained against the group-theoretic definitions and does not rely on self-citation chains or ansatzes smuggled from prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of group theory together with the definition of the χ operator introduced by Sidki; no new entities or free parameters are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of group theory (associativity, inverses, identity)
    The paper works entirely inside the category of groups.
  • domain assumption Definition of the weak commutativity operator χ as given by Sidki
    The construction is taken as given from prior literature.

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Reference graph

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