Uniform Cyclic Group Factorizations of Finite Groups

Abstract

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 $H_1,H_2,\dots ,H_k$ 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.

10 pages. To appear in Communications in Algebra