发表机构
Dalhousie University（达尔豪斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Forcade和Pollington1990年提出的关于初始区间上群映射的猜想，即当S为{1,2,…,k}时，大小为k的群G必为阿贝尔群。
AI 中文摘要
设S为整数集ℤ的一个k元子集，G为大小为k的乘法群，假设存在单射函数φ:S→G满足：若x,y,z∈S且z=xy，则φ(z)=φ(x)φ(y)。1990年，Forcade和Pollington猜想当S={1,2,…,k}时，G是阿贝尔群，该猜想被Richard K. Guy列为未解决问题，2021年Caicedo等人在《Electron. J. Combin.》中仍指出其未解决。我们成功证明了Forcade-Pollington猜想在一般情况下成立。
英文摘要
Let $S$ denote a $k$-subset of $\mathbb{Z}$. Let $G$ be a (multiplicative) group of size $k$. Assume that there exists a function $φ\colon S \to G$ that is injective and that satisfies the following: If $x, y, z \in S$ and $z = xy$, then $φ(z) = φ(x) φ(y)$. 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.
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