This writing assignment is written in fulfillment for the Abstract Algebra course in the Spring 2022 semester.

The Fundamental Theorem of Finite Abelian Groups states that any finite abelian group can be written as the direct product of cyclic subgroups. This reduces most questions about finite abelian groups to questions about cyclic groups. The results on the structure of finite abelian groups are really special cases of some wider and deeper theorems. Accordingly, a group can be written as the direct product of its so-called P-Sylow subgroups. An abelian group has one p-Sylow subgroup for every prime p dividing its order. For each prime P which is a divisor of the order of the group there are as many groups of order $p^{a_i}$ as there are partitions of $a_i$. So the number of nonisomorphic abelian groups of order $n = p^{a_1} ... p^{a_k}$ is $f(a_1)f(a_2) .. · f(a_k)$, where $f(m)$ denotes the number of partitions of m. Thus we know how many nonisomorphic finite abelian groups there are for any given order; hence the focus of this project.

In this paper, our goal is to apply the fundamental theorem of finite abelian groups on a defined abelian group. The group we are using in this paper is: U(432). U(432) is the group of all classes, each of which is a positive number smaller than 432 and relatively prime to 432,i.e.U(432) is the group of all positive i’s smaller than 432 where gcdi432 = 1 (gcd is the abbreviation of Greastest Common Divisor). Furthermore, U(432)is a group under the operation multiplication modulo 432. Having our group defined, in sections 2 & 3, we will first introduce the underlying theories of the fundamental theorem of finite abelian groups. We will then use our predefined group to put the fundamental theorem of finite abelian groups to practice. The process is done by first finding the order of the group and subsequently finding the elements of U(432)(section 4). Accordingly, in section 5, we indicate how many isomorphisms are there possible for U(432). In section 6 we find the order of all elements in U(432)and write them in table 1. By looking at table 1 and the orders of the elements, we can cancel out the unaccepted isomorphisms. Moreover, in section 7 we find the distinctive cyclic subgroups generated by the elements of U(432). Finally in section 8, we will show examples of how to write U(432)as the direct product of it’s cyclic sbgroups. For the coding sections, "python" has been used.