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2013 (Vol. 5, Issue: 06)
Article Information:

Q (√m)/Q Under the Action of PSL_2 (Z)∩〈x,y: x^2= y^6=1〉

M. Aslam Malik and M. Asim Zafar
Corresponding Author:  M. Aslam 

Key words:  G-set, legendre symbol, linear fractional transformations, , , ,
Vol. 5 , (06): 1916-1922
Submitted Accepted Published
June 07, 2012 July 09, 2012 February 21, 2013

This study is concerned with the natural action (as Möbius transformations) of some subgroups of PGL2(Z) on the elements of quadratic number field over the rational numbers. We start with two groups- the full modular group G = PSL2(Z) and another group of Möbius transformations M= 〈x,y: x^2= y^6=1〉. 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.
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  Cite this Reference:
M. Aslam Malik and M. Asim Zafar, 2013. Q (√m)/Q Under the Action of PSL_2 (Z)∩〈x,y: x^2= y^6=1〉.  Research Journal of Applied Sciences, Engineering and Technology, 5(06): 1916-1922.
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ISSN (Online):  2040-7467
ISSN (Print):   2040-7459
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