Uniform Cyclic Group Factorizations of Finite Groups

TL;DR

This paper introduces the concept of uniform group factorizations, generalizes exact factorizations, and shows that all finite solvable groups admit uniform cyclic factorizations, linking the problem to simple groups.

Contribution

It defines uniform group factorizations, proves all finite solvable groups have uniform cyclic factorizations, and relates the existence of such factorizations to simple groups.

Findings

All finite solvable groups admit uniform cyclic factorizations

The existence of uniform cyclic factorizations for all finite groups is linked to simple groups

Concrete examples of uniform cyclic group factorizations are provided

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 $H_1, H_2, \ldots, H_k$ 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.