eng
Ansari Education and Research Society
Journal of Ultra Scientist of Physical Sciences
December 2019
31
12
106
114
http://dx.doi.org/10.22147/jusps-A/311201
1522
article
Existence of Smooth Epimorphism from a Fuchsian Group to the point Group of Sulfur-Hexafloride
MOLOYA BHUYAN
1
CHANDRA CHUTIA
2
Department of Mathematics, DCB Girlsu2019 College, Jorhat-ASSAM (India)
Department of Mathematics, Jorhat Institute of Science & Technology, Jorhat-ASSAM (India)
<p style="text-align:justify">The theory of Fuchsian group plays an important role in the study of compact Riemann surfaces Automorphism groups, which was initially studied by A.M. Macbeth. The biholomorphic self transformations of a compact Riemann surface S of genus g (<sub><img alt="\small \geq" src="http://latex.codecogs.com/gif.latex?%5Csmall%20%5Cgeq" /></sub>2) forms a finite group whose order cannot exceed 84 (g -1). This maximum bound is attained for infinitely many values of g , the least being 3. The groups for which this bound is attained are called Hurwitz groups, and this groups belong to the class of perfect groups which are non soluble. In context to the class of soluble groups, the corresponding bound is 48 (g -1) and this bound is also attained for infinitely many values of g . The problem of finding such bounds for various sub-classes of the finite soluble groups and the number of values of g for which these bounds are attained has been the theme of many research papers during the last few decades.</p>
<p style="text-align:justify">In this paper, a set of necessary and sufficient conditions for the existence of smooth epimorphism from a Fuchsian group to the point group O<sub>h</sub>, which belongs to the sub-class of octahedral group considering the symmetries of Sulfur-Hexafloride ( S F<sub>6</sub> ) molecule Having the abstract group representation <img alt="\small \left \langle \alpha" src="http://latex.codecogs.com/gif.latex?%5Csmall%20%5Cleft%20%5Clangle%20%5Calpha" /><sup>2</sup>=<img alt="\beta" src="http://latex.codecogs.com/gif.latex?%5Cbeta" /><sup> 4</sup>=(<img alt="\alpha \beta" src="http://latex.codecogs.com/gif.latex?%5Calpha%20%5Cbeta" />) <sup>2</sup>=1<img alt="\rangle \right \rangle" src="http://latex.codecogs.com/gif.latex?%5Crangle%20%5Cright%20%5Crangle" /> is established to fulfill the objective.</p>
http://www.ultrascientist.org/paper/1522/
Compact Riemann Surface
Fuchsian group
Molecular Symmetries
Smooth Epimorphism
1991 Mathematics Subject Classification: 20H10, 30F10