Year: 2026
Type: preprint
Abstract: Let $S$ denote a $k$-subset of $\mathbb{Z}$. Let $G$ be a (multiplicative) group of size $k$. Assume that there exists a function $\varphi\colon S \to G$ that is injective and that satisfies the following: If $x, y, z \in S$ and $z = xy$, then $\varphi(z) = \varphi(x) \varphi(y)$. In 1990, Forcade and Pollington conjectured that if $S = \{ 1, 2, \ldots, k \}$, 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.
Source: arXiv (Cornell University)
Author: Campbell, John M.
Institution: Dalhousie University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207836346