IJPAM: Volume 79, No. 3 (2012)

FINITE ABELIAN GROUPS BASED ON JR-2CN

Rand Alfaris$^1$, Hailiza Kamarulhaili$^2$
$^{1,2}$School of Mathematical Sciences
Universiti Sains Malaysia
11800USM, Penang, MALAYSIA


Abstract. This paper is concerned with establishing new type of finite Abelian groups based on the infinite Abelian group JR-2CN. JR-2CN is the set of infinite integer numbers that can be represented as a summation of two signed cubic numbers. We designate each finite Abelian group generated from JR-2CN as 2JR$_n$, where $n$ is used to determine the order of the group 2JR$_n$. The addition binary operations that are applied to construct these finite Abelian groups are originally based on addition operation of JR-2CN, but under the concept of the arithmetic modulo. Since the elements of JR-2CN are ordered pairs, therefore, it is crucial to apply the modulo on each component for each ordered pair. Theorems and propositions related to three essential parts of this study are stated, and proved. The first part is determining the nature of the elements. The second part is concerned with the critical status of the order of each set of 2JR$_n$, and last part is the Abelian group proof.

Received: June 30, 2012

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

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

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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: 3