On the number of exact factorization of finite Groups

TL;DR

This paper investigates the function counting exact factorizations of finite groups, providing explicit calculations for certain groups and asymptotic estimates for alternating groups, while proposing open questions for future research.

Contribution

It introduces the function $f_2(G)$ for counting exact factorizations, computes it for specific groups, and derives asymptotic formulas for alternating groups, advancing understanding of group factorizations.

Findings

Computed $f_2(G)$ for well-known finite groups

Derived asymptotic expression for $f_2(A_{2^n})$

Proposed open questions for further research

Abstract

In this work, we study the function $f_2(G)$ that counts the number of exact factorizations of a finite group $G$. We compute $f_2(G)$ for some well-known families of finite groups and use the results of Wiegold and Williamson \cite{WW} to derive an asymptotic expression for the number of exact factorizations of the alternating group $A_{2^n}$. Finally, we propose several questions about the function $f_2(G)$ that may be of interest for further research.