有限群的均匀循环群分解
Uniform Cyclic Group Factorizations of Finite Groups
AI总结：
本文提出均匀群分解概念，证明有限可解群均有均匀循环群分解，并揭示该问题与有限单群均匀分解的等价性，附有实例。
摘要：
本文引入了一种有限群的分解类型，称为均匀群分解，作为有限群精确分解的推广。若存在子群 $H_1, H_2, \ldots, H_k$，使得 $G = H_1 H_2 \cdots H_k$，且将任意元素 $g \in G$ 表示为 $g = h_1 h_2 \cdots h_k$（其中 $h_i \in H_i$）的方式数目不依赖于 $g$ 的选择，则称群 $G$ 允许均匀群分解。此外，由循环子群构成的均匀群分解称为均匀循环群分解。首先，我们证明任何有限可解群都允许均匀循环群分解。其次，我们证明所有有限群是否允许均匀循环群分解等价于所有有限单群是否允许均匀群分解。最后，我们给出此类分解的一些具体例子。
英文摘要：
In this paper, we introduce a kind of decomposition of a finite group called a uniform group factorization, as a generalization of exact factorizations of a finite group. A group $G$ is said to admit a uniform group factorization if there exist subgroups $H_1, H_2, \ldots, H_k$ such that $G = H_1 H_2 \cdots H_k$ and the number of ways to represent any element $g \in G$ as $g = h_1 h_2 \cdots h_k$ ($h_i \in H_i$) does not depend on the choice of $g$. Moreover, a uniform group factorization consisting of cyclic subgroups is called a uniform cyclic group factorization. First, we show that any finite solvable group admits a uniform cyclic group factorization. Second, we show that whether all finite groups admit uniform cyclic group factorizations or not is equivalent to whether all finite simple groups admit uniform group factorizations or not. Lastly, we give some concrete examples of such factorizations.
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。
