Counting Distinct Fuzzy Subgroups of Finite Abelian Groups of Order 𝑝𝑛𝑞𝑚
Lingling Han and Xiuyun Guo
The purpose of this paper is to count the distinct fuzzy subgroups of a finite abelian group of order 𝑝𝑛𝑞𝑚 for any different primes 𝑝, 𝑞 and any positive integers 𝑛, 𝑚. This counting problem is reduced to finite anelian 𝑝-groups. As applications of our main result, explicit formulas for the number of distinct fuzzy subgroups of the following two classes of finite abelian groups are given:
i) The direct product ℤ𝑛𝑝 × ℤ𝑚𝑞 of a finite elementary abelian 𝑝-group ℤ𝑛𝑝 and a finite elementary abelian 𝑞-group ℤ𝑚𝑞 with different primes 𝑝 and 𝑞;
ii) The direct product ℤ𝑝𝑛 × ℤ𝑚𝑞 of a finite cyclic 𝑝-group ℤ𝑝𝑛 and a finite elementary abelian 𝑞-group ℤ𝑚𝑞 with different primes 𝑝 and 𝑞.
Keywords: Fuzzy subgroup, finite abelian group, subgroup chain, cyclic group, elementary abelian 𝑝-group, direct product