# IJPAM: Volume 79, No. 4 (2012)

**FINITE ABELIAN GROUPS BASED ON JR-3CN**

School of Mathematical Sciences

Universiti Sains Malaysia

11800USM, Penang, MALAYSIA

**Abstract. **A new type of finite Abelian groups, whose elements are the integer numbers that can be represented as a sum of three signed cubes, is constructed in this paper. These integers are the elements of three signed cubes sum family that has been introduced in JR-3CN family. Following the definition of this family, the representation of each element in the infinite Abelian groups is as an ordered pair, whose components and its subscribed parameter are the three signed cubes. The finite Abelian groups are constructed here, considered as a continuum to the study that has been done on 2JR. Each finite Abelian group is denoted by 3JR, where, refers to the three signed cubes and refers to the modulo and determines the order of the group. An addition binary operation has been defined on 3JR based on the addition binary operation associated with JR-3CN. This addition binary operation is operated under modulo . Since the structure of the integers in JR-3CN is represented as ordered pairs, therefore, it is decisive to apply the modulo on each component for each ordered pair whenever it is needed. In order to support the claim, theorems and propositions on constructing the finite sets and the finite Abelian groups 3JR are stated and proved. Where they are mainly concern with determining the type of the elements and the order of each finite set.

**Received: **June 17, 2012

**AMS Subject Classification: **11R16, 20C07, 14H10

**Key Words and Phrases: **cubic numbers, integer representations of sum of three cubic numbers, Abelian groups JR-2CN and JR-3CN, Finite Abelian groups 2JR

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**Source:**International Journal of Pure and Applied Mathematics

**ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2012

**Volume:**79

**Issue:**4