On the factorization numbers of some finite $p$-groups

TL;DR

This paper computes the factorization number of certain finite abelian groups using Möbius inversion, providing explicit formulas that improve previous results in the literature.

Contribution

It introduces explicit formulas for the factorization number of specific finite abelian groups, advancing the understanding of their algebraic structure.

Findings

Derived explicit formulas for $F_2(G)$ for two classes of finite abelian groups.

Improved upon previous results by providing more general and precise expressions.

Enhanced the computational methods for factorization numbers in finite group theory.

Abstract

This note deals with the computation of the factorization number $F_2(G)$ of a finite group $G$. By using the M\"{o}bius inversion formula, explicit expressions of $F_2(G)$ are obtained for two classes of finite abelian groups, improving the results of {\it Factorization numbers of some finite groups}, Glasgow Math. J. (2012).