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Counting Distinct Fuzzy Subgroups of Finite Abelian Groups of Order π‘π‘›π‘žπ‘š
Lingling Han and Xiuyun Guo

The purpose of this paper is to count the distinct fuzzy subgroups of a finite abelian group of order π‘π‘›π‘žπ‘š for any different primes 𝑝, π‘ž and any positive integers 𝑛, π‘š. 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 ℀𝑛𝑝 Γ— β„€π‘šπ‘ž of a finite elementary abelian 𝑝-group ℀𝑛𝑝 and a finite elementary abelian π‘ž-group β„€π‘šπ‘žΒ with different primes 𝑝 and π‘ž;

ii) The direct product ℀𝑝𝑛 Γ— β„€π‘šπ‘ž of a finite cyclic 𝑝-group ℀𝑝𝑛 and a finite elementary abelian π‘ž-group β„€π‘šπ‘ž with different primes 𝑝 and π‘ž.

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

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