JUMLAH GRUP BAGIAN DALAM DARAB LANGSUNG GRUP SIKLIS BERHINGGA

Abstract

The problem that will be investigated in this study is the answer of the question on the number of subgroups in a direct product of finite cyclic groups. The answer of this question is generally not easy. A number of authors have counted subgroups in certain classes of finite groups. Joseph Petrillo in his paper entitled Counting Subgroups in a Direct Product of Finite Cyclic Groups, in The College Mathematics Journal, Vol.42, No.3, 2011 contributed his idea to the case of a direct product of finite cyclic groups. This researh is a literature study of the Joseph Petrillos paper.

For a finite group G with subgroup lattice L(G), let | L(G)| denote the number of subgroups of G. Let Zn denote the unique cyclic group of order n, which can be viewed as the group of integers under adition modulo n. Our goal is to derive a formula for calculating |L(ZmxZn)| for all positive integers m and n. The main tool is a theorem of Goursat, which we introduce next. First, we analyze the cases where m and n are relatively prime and powers of the same prime. Then, we extend the results to the direct product of arbitrary cyclic and non-cyclic groups.

Key words: Group, Cyclic group, Order of a grup, Goursats theorem.