Counting the number of fuzzy subgroups of an abelian group of order p^nq^m

In this paper, we determine the number of fuzzy subgroups of G=Z p^n + Z q^m where p,q are distinct primes, and n,m are any two natural numbers. First, we represent the number of maximal chains of G as a summation through m involving certain binomial coefficients. Using this concise expression, we propose a combinatorial formula for the number of fuzzy subgroups of G and verify it for specific values of m and n. Finally, we sketch a proof for the general case.