论文标题
通过通勤子集进行分区的组
Groups that have a Partition by Commuting Subsets
论文作者
论文摘要
令$ g $为非亚伯人群。我们说$ g $具有Abelian分区，如果存在$ g $的分区，以通勤子集$ a_1，a_2，\ ldots，a_n $ of $ g $，以便每个$ | a_i | \ geqslant 2 $ for $ i = 1，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，2，\ ldots，n $。本文调查了与Abelian分区的小组有关的问题。除其他结果外，我们还表明，每个有限组都是具有Abelian分区的组的子组的同构，并且也与没有Abelian分区的组的亚组同构。我们还为几个群体家庭的分区数量最小数量的界限 - 在某些情况下进行了确切的计算。最后，我们研究了与直接产品相对于直接产品的最小零件数量的分区大小。
Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper investigates problems relating to group with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions -- with exact calculations in some cases. Finally, we examine how the size of partitions with the minimum number of parts behaves with respect to the direct product.
