关于Street和Whitehead关于局部极大无积集的一个猜想
On a conjecture of Street and Whitehead on locally maximal product-free sets
摘要：
设$S$是群$G$的非空子集。如果$S\cap SS=\varnothing$是，则我们说$S$是无积的，如果只要$T$是无积的，并且$S\subseteq T$，则$S=T$是局部最大的。最后，如果$G^*\subseteq S \sqcup SS$（其中$G^*$是$G$的所有非单位元素的集合），则$S$填充$G$，并且如果$G$中的每个局部极大无积集填充$G$，则$G$是填充群。Street和Whitehead（在“Group Ramsey Theory”，J. Comb. Theory Series A，17（1974）219-226中）研究了填充群，并给出了填充阿贝尔群的分类。本文得到了非交换情形下的填充群的一些结果，包括奇阶填充群的一个分类。Street和Whitehead证明了当$n=6k+1$（$k\geq 1$）时，$2n$阶的有限二面体群不被填充.我们在二面体群上证明了这个猜想是错误的，并由此得到了二面体群中大小为3和4的局部极大无积集的分类。
英文摘要：
Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing$, and $S$ is locally maximal if whenever $T$ is product-free and $S\subseteq T$, then $S=T$. Finally $S$ fills $G$ if $G^*\subseteq S \sqcup SS$ (where $G^*$ is the set of all non-identity elements of $G$), and $G$ is a filled group if every locally maximal product-free set in $G$ fills $G$. Street and Whitehead (in `Group Ramsey Theory', J. Comb. Theory Series A, 17 (1974) 219-226) investigated filled groups and gave a classification of filled abelian groups. In this paper, we obtain some results about filled groups in the non-abelian case, including a classification of filled groups of odd order. Street and Whitehead conjectured that the finite dihedral group of order $2n$ is not filled when $n=6k+1$ ($k\geq 1$). We disprove this conjecture on dihedral groups, and in doing so obtain a classification of locally maximal product-free sets of sizes 3 and 4 in dihedral groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
