On the number of exact factorization of finite Groups

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.