Group maps on initial intervals — Draconitic Research

Abstract

Let S denote a k-subset of ℤ. Let G be a (multiplicative) group of size k. Assume that there exists a function φ: S → G that is injective and that satisfies the following: If x, y, z ∈ S and z = xy, then φ(z) = φ(x) φ(y). In 1990, Forcade and Pollington conjectured that if S = { 1, 2, …, k }, then G is abelian. This has been a long-standing conjecture highlighted by Richard K. Guy as an unsolved problem, and this problem appears to have remained open, with its open status having been noted by Caicedo et al. in 2021 [Electron. J. Combin.]. We succeed in proving the Forcade-Pollington conjecture in the affirmative and in full generality.

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BibTeX RIS

BibTeX

@misc{campbell2026group,
  title = {{Group maps on initial intervals}},
  author = {John M. Campbell},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12087},
  eprint = {2610.12087},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12087},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Group maps on initial intervals
AU  - John M. Campbell
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12087
UR  - https://arxiv.org/abs/2610.12087
AB  - Let S denote a k-subset of ℤ. Let G be a (multiplicative) group of size k. Assume that there exists a function φ: S → G that is injective and that satisfies the following: If x, y, z ∈ S and z = xy, then φ(z) = φ(x) φ(y). In 1990, Forcade and Pollington conjectured that if S = { 1, 2, …, k }, then G is abelian. This has been a long-standing conjecture highlighted by Richard K. Guy as an unsolved problem, and this problem appears to have remained open, with its open status having been noted by Caicedo et al. in 2021 [Electron. J. Combin.]. We succeed in proving the Forcade-Pollington conjecture in the affirmative and in full generality.
ER  -

Abstract

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