On the Number of Fuzzy Subgroups of Finite Symmetric Groups

One of the most important problems of fuzzy group theory is to classify the fuzzy subgroups of a finite group. This topic has enjoyed a rapid development in the last few years. Several papers have treated the particular case of finite abelian groups. Thus, in [13] the number of distinct fuzzy subgroups of a finite cyclic group of square-free order is determined, while [14–16] and [28] deal with this number for cyclic groups of order pnqm (p, q primes). Also, recall here the paper [25], where a recurrence relation is indicated which can successfully be used to count the number of distinct fuzzy subgroups for two classes of finite abelian groups: (arbitrary) finite cyclic groups and finite elementary abelian p-groups. The explicit formula obtained for the first class leads in [22] to an expression of the well-known central Delannoy numbers. Other related works on this topic are [1, 3, 4, 17, 24].