International J.Math. Combin. Vol.4(2022), 32-40 The Number of Chains of Subgroups in the Lattice of Subgroups of Group Z m × A n ,n ≤ 6,m ≤ 3 Mike Ogiugo 1 , Amit Seghal 2 , S. A. Adebisi 3 and M. EniOluwafe 4 1. Department of Mathematics,School of Science, Yaba College of Technology, Nigeria 2. Department of Mathematics, Pt. Neki Ram Sharma Government College, Rohtak, India 3. Department of Mathematics, Faculty of Science, University of Lagos, Nigeria 4. Department of Mathematics, Faculty of Science, University of Ibadan, Nigeria E-mail: ekpenogiugo@gmail.com, amit sehgal iit@yahoo.com adesinasunday@yahoo.com, michael.enioluwafe@gmail.com Abstract: In this paper, we established the number of chains of subgroups in the subgroup lattice of the Cartesian product of the alternating group and cyclic group using computational technique induced by the set of representatives of isomorphism classes of subgroups. Key Words: Subgroups, alternating Group, chains of subgroups and fuzzy subgroups. AMS(2010): 20D06, 20D30, 20E15. §1. Introduction All groups are finite throughout this paper.For a given group G, the lattice of subgroups is (L(G), ≤) here by L(G) mean all subgroups of G and the partial order ≤ works as set inclusion. If chain of subgroups of G contains G then it is called rooted (more precisely G-rooted) otherwise it is known as unrooted. In this paper the study of chains of subgroups describes the set containing all chains of subgroups of G, which ends in with G.A formula of lattice of a finite cyclic group, for number of chains of subgroups was given by Tˇ arnˇ auceanu and Bentea [1] by giving its one variable generating function. J.M. Oh in his paper [2] determined the number of subgroups of a finite cyclic group of 4n by giving its multi variables generating function. The problem of counting chains of subgroups in the lattice of subgroups of for any given group G got attention of researchers specially classifying fuzzy subgroups of G under a specific type of equivalence relation (see [3], [4]). §2. Preliminaries A set of subgroups of G fully ordered by set inclusion is a chain of subgroups of group G. In this paper, the chain of subgroups of G which end in G. 1 Received November 15, 2022, Accepted December 6, 2022.