55 Journal of Advances in Computer Research Quarterly ISSN: 2008-6148 Sari Branch, Islamic Azad University, Sari, I.R.Iran (Vol. 4, No. 1, February 2013), Pages: 55-63 www.jacr.iausari.ac.ir A Recursive Formula for the Number of Fuzzy Subgroups of Finite Cyclic Groups Mahdi Imanparast 1* , Hamid Darabi 2 (1) Department of Computer Science, University of Bojnord, Bojnord, Iran (2) Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran m.imanparast@ub.ac.ir; darabi@iauesf.ac.ir Received: 2012/10/03; Accepted: 2012/12/19 Abstract Fuzzy groups and fuzzy theory have a lot of applications in several sciences such as mathematics, computer science, computer and electrical engineering. Hence, counting the number of fuzzy subgroups of finite groups to classify them is an important issue in fuzzy theory. The Main goal of this paper is to give an explicit formula for the number of fuzzy subgroups of a finite cyclic group 1 2 3 k p p p p G Z Z Z Z = · · · ·  , where k p p p ,..., , 2 1 are distinct prime numbers. We introduce a very simple recursive formula to count the number of subgroups of G. Keywords: Fuzzy groups, Lattice, Finite cyclic groups, Chains 1. Introduction One of the most important problems of fuzzy group theory is to classify the fuzzy subgroups of a finite group [1]. Several papers have treated the particular case of finite abelian group. Laszlo [2] studied the construction of fuzzy subgroups of groups of the orders one to six. Zhang and Zou [3] have determined the number of fuzzy subgroups of cyclic groups of the order n p where p is a prime number. Murali and Makamba in [4] and [5], considering a similar problem, found the number of fuzzy subgroups of abelian groups of the order m n q p where p and q are different primes. In [6], Tarnauceanu and Bentea established the recurrence relation verified by the number of fuzzy subgroups of finite cyclic groups. Their result is the improving of Murali ’s works in [4] and [5]. Finally, authors in [8] establish an idea to count the number of fuzzy subgroups in the particular case of finite cyclic groups, namely s r q p Z Z Z Z · · · where , , pqr and s are distinct prime numbers. Finding the number of fuzzy subgroups is a common problem in fuzzy groups to classify them. This paper discusses a particular case of finite cyclic groups k p p p Z Z Z G · · · =  2 1 , where k p p p ,..., , 2 1 are distinct prime numbers. However, the result is a generalization of [8] approach in a general case.