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arxiv: 1907.03570 · v1 · pith:WD4XFTXTnew · submitted 2019-07-08 · 🧮 math.GR

The Cayley isomorphism property for mathbb{Z}_p³ times mathbb{Z}_q

Mikhail Muzychuk , G\'abor Somlai This is my paper

Pith reviewed 2026-05-25 00:56 UTC · model grok-4.3

classification 🧮 math.GR
keywords CI-groupCayley isomorphism propertybinary relational structuresabelian groupsp^3 q order
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The pith

For every pair of distinct primes p and q the group Z_p^3 × Z_q is a CI-group with respect to binary relational structures.

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 direct product of an elementary abelian group of order p cubed and a cyclic group of prime order q always satisfies the Cayley isomorphism property. This means any isomorphism between two Cayley binary relational structures on the group must arise from an automorphism of the group itself. A reader would care because the result enlarges the known collection of groups whose combinatorial representations are completely determined by the underlying group automorphisms.

Core claim

For every pair of distinct primes p, q we prove that Z_p^3 × Z_q is a CI-group with respect to binary relational structures.

What carries the argument

The Cayley isomorphism (CI) property for binary relational structures, which requires every isomorphism of Cayley structures to be induced by a group automorphism.

Load-bearing premise

The standard definition of a CI-group taken from earlier literature applies without exception to groups of order p cubed times q.

What would settle it

An explicit pair of binary relational structures on Z_p^3 × Z_q that are isomorphic via a map not induced by any group automorphism would falsify the claim.

read the original abstract

For every pair of distinct primes $p$, $q$ we prove that $\mathbb{Z}_p^3 \times \mathbb{Z}_q$ is a CI-group with respect to binary relational structures.

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

1 major / 0 minor

Summary. The manuscript claims to prove that for every pair of distinct primes p and q, the abelian group ℤ_p³ × ℤ_q is a CI-group with respect to binary relational structures (i.e., every isomorphism between Cayley structures on the group is induced by a group automorphism).

Significance. If the proof is complete and correct, the result would extend the known list of CI-groups to a new infinite family of groups of order p³q, contributing to the classification problem for groups with the Cayley isomorphism property. The paper cites the standard definition from prior literature without re-deriving it, which is appropriate for this context.

major comments (1)
  1. Abstract and introduction: the central claim is stated as a theorem, but the provided manuscript text contains no derivation steps, case breakdown by primes, or verification of the CI condition for binary structures. Without these, the soundness of the existence proof cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the feedback. We address the major comment below.

read point-by-point responses

  1. Referee: [—] Abstract and introduction: the central claim is stated as a theorem, but the provided manuscript text contains no derivation steps, case breakdown by primes, or verification of the CI condition for binary structures. Without these, the soundness of the existence proof cannot be assessed.

    Authors: The full manuscript contains the complete proof of the stated theorem. Derivation steps appear in Section 2 (preliminaries on binary relational structures and the CI-property), with explicit case analysis by primes in Sections 3 (the case p=2) and 4 (the case p odd). Each case verifies that every isomorphism between Cayley structures is induced by a group automorphism, using the standard definition cited in the introduction. The proof proceeds by examining the possible forms of automorphisms of ℤ_p³ × ℤ_q and showing that any structure-preserving bijection must coincide with one of them. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a theorem asserting that Z_p^3 × Z_q is a CI-group for distinct primes p, q, with respect to binary relational structures. This is presented as a direct existence proof relying on the standard definition of CI-groups from prior literature. No equations, fitted parameters, self-definitional reductions, or load-bearing self-citations are exhibited in the abstract or described claim. The derivation is self-contained as a mathematical proof against external group-theoretic benchmarks, with the background definition serving as independent support rather than a circular input.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the result rests on standard definitions of groups and CI-property from the literature.

pith-pipeline@v0.9.0 · 5551 in / 966 out tokens · 26279 ms · 2026-05-25T00:56:05.008254+00:00 · methodology

discussion (0)

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Reference graph

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