MVLSC 35.3-4, p. 263-280

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Counting Distinct Fuzzy Subgroups of Finite Abelian Groups of Order 𝑝^𝑛𝑞^𝑚
Lingling Han and Xiuyun Guo

The purpose of this paper is to count the distinct fuzzy subgroups of a finite abelian group of order 𝑝^𝑛𝑞^𝑚 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 ℤ^𝑛_𝑝 × ℤ^𝑚_𝑞 of a finite elementary abelian 𝑝-group ℤ^𝑛_𝑝 and a finite elementary abelian 𝑞-group ℤ^𝑚_𝑞 with different primes 𝑝 and 𝑞;

ii) The direct product ℤ^𝑝_𝑛 × ℤ^𝑚_𝑞 of a finite cyclic 𝑝-group ℤ^𝑝_𝑛 and a finite elementary abelian 𝑞-group ℤ^𝑚_𝑞 with different primes 𝑝 and 𝑞.

Keywords: Fuzzy subgroup, finite abelian group, subgroup chain, cyclic group, elementary abelian 𝑝-group, direct product

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