IJPAM: Volume 116, No. 3 (2017)

Title

ORDER DIMENSIONS OF THE DIAGONAL MATRIX
GROUP AND CERTAIN SUBGROUPS OF
THE SPECIAL LINEAR GROUP

Authors

A. Putmuang$^1$, J. Sukultanasorn$^2$, M. Klubmungmee$^3$, N. Sirasuntorn$^{4}$
$^{1,2,3,4}$Department of Mathematics
Faculty of Science
Srinakharinwirot University
114 Sukhumvit 23, Wattana District, Bangkok 10110, THAILAND

Abstract

Let $G$ be a group of order $m$. An order dimension of $G$ is the number of different orders of nonidentity elements of $G$ denoted by odim($G$). Let $m$ be a product of prime powers, that is, $m$ = ${p_1}^{m_1} {p_2}^{m_2} \cdots {p_k}^{m_k}$ where $p_1, p_2,\ldots, p_k$ are distinct primes and $m_1$ is the greatest of all the powers, i.e. $m_1\geq m_i$ for all $i\in \{2, 3, \ldots, k\}$. We say that the group $G$ has $property$ # if $G$ has a proper subgroup $H$ that odim($H$)= $m_1(m_2+1)\cdots(m_k+1)-1$.

In this paper, we study order dimensions and the property # of the diagonal matrix group and certain subgroups of the special linear group over a finite field.

History

Received: 2017-06-27
Revised: 2017-09-02
Published: October 26, 2017

AMS Classification, Key Words

AMS Subject Classification: 20H30
Key Words and Phrases: order dimension, diagonal matrix group, special linear group

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Bibliography

1
I. Ganev, Order Dimension of Subgroups, Rose-Hulman Undergraduate Mathematics Journal, 9, No. 2 (2008), 1-10.

2
J. A. Gallian, Contemporary Abstract Algebra (Eighth Edition), Cengage Learning, USA (2013).

3
N. I. Herstein, Abstract Algebra (Second Edition), Collier Macmillan Publishers, England (1990).

How to Cite?

DOI: 10.12732/ijpam.v116i3.19 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 116
Issue: 3
Pages: 727 - 737


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