Research Article | OPEN ACCESS
$Q(√m)Q$ Under the Action of $PSL_2(Z)∩ \langle x, y : x^2 = y^6 = 1\rangle$
M. Aslam Malik and M. Asim Zafar
Department of Mathematics, University of the Punjab Quaid-e-Azam Campus, Lahore-54590, Pakistan
Research Journal of Applied Sciences, Engineering and Technology 2013 6:1916-1922
Received: June 07, 2012 | Accepted: July 09, 2012 | Published: February 21, 2013
Abstract
This study is concerned with the natural action (as Möbius transformations) of some subgroups of $PGL_2 (Ζ)$ on the elements of quadratic number field over the rational numbers. We start with two groups- the full modular group $G = PSL_2 (Ζ)$ and another group of Möbius transformations $M = \langle x, y : x^2 = y^6 = 1\rangle$. We consider different sets of numbers with fixed discriminants in the quadratic field and look at structure of the orbits orbits of the actions of $G , M , G ∩ M$ and their subgroups on these sets. The results of earlier studies on the number of orbits and the properties of elements belonging to them are extended by similar results related to the new twist connected to the group M which has nontrivial intersection with G and opens a possibility to look at orbits which were not computed in earlier studies.
Keywords:
G-set, legendre symbol, linear fractional transformations,
Competing interests
The authors have no competing interests.
Open Access Policy
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Copyright
The authors have no competing interests.
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ISSN (Online): 2040-7467
ISSN (Print): 2040-7459 |
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