Factorizations of simple groups of order 168 and 360

TL;DR

The paper investigates multifold-factorizability of finite groups, focusing on the simple groups of orders $168$ and $360$ by constructing explicit factorizations aligned with all prime decompositions of these orders. It employs subgroup structure (such as $S_4$ and $C_7\rtimes C_3$ in the $168$-order group, and $A_5$, $S_4$, and other subgroups in the $360$-order group) together with computational tools (GAP and GRAPE) to realize all required factorizations, along with general ABCD-decomposition techniques and reductions via transversals and double cosets. The key contributions include proving both simple groups are multifold-factorizable and outlining scalable methods for finding such factorizations in other groups, as well as proposing conjectures that broaden the scope to $S_n$ and $A_n$. These results deepen our understanding of when finite groups admit factorizations for all admissible multiplier sets and suggest structural criteria linking subgroup arrangements to multifold-factorizability.

Abstract

A finite group $G$ is called $k$-factorizable if for any factorization $|G|=a_1\cdots a_k$ with $a_i>1$ there exist subsets $A_i$ of $G$ with $|A_i|=a_i$ such that $G=A_1\cdots A_k$. We say that $G$ is \textit{multifold-factorizable} if $G$ is $k$-factorizable for any possible integer $k\geq2$. We prove that simple groups of orders 168 and 360 are multifold-factorizable and formulate two conjectures that the symmetric group $S_n$ for any $n$ and the alternative group $A_n$ for $n\geq6$ are multifold-factorizable.

Theorems & Definitions (30)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Corollary 2.4
• Lemma 2.5
• Lemma 2.6
• Corollary 2.7
• Corollary 2.8
• Lemma 2.9
• Corollary 2.10