An explicit formula for the number of fuzzy subgroups of a finite abelian $p$-group of rank two

Abstract

Ngcibi, Murali and Makamba [Fuzzy subgroups of rank two abelian<br />$p$-group, Iranian J. of Fuzzy Systems {bf 7} (2010), 149-153]<br />considered the number of fuzzy subgroups of a finite abelian<br />$p$-group $mathbb{Z}_{p^m}times mathbb{Z}_{p^n}$ of rank two, and<br />gave explicit formulas for the cases when $m$ is any positive<br />integer and $n=1,2,3$. Even though their method can be used for the<br />cases when $n=4,5,ldots$ by using inductive arguments, it does not<br />provide an explicit formula for that number for an arbitrarily<br />given positive integer $n$. In this paper we give a complete answer<br />to this problem. Thus for arbitrarily given positive integers $m$<br />and $n$, an explicit formula for the number of fuzzy subgroups of<br />$mathbb{Z}_{p^m}times mathbb{Z}_{p^n}$ is given.