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A finite group G acts as a group of automorphisms on a compact Riemann surface S of genus g if and only if there exist a Fuchsian group Φ and an epimorphism Φ: Γ→G such that kerΦ = K is a surface group of genus g. And then Φ is named as smooth homomorphism. The objective of this paper is to establish a set of necessary and sufficient conditions for the existence of smooth epimorphism from a Fuchsian group to the finite group of symmetries of Carbon Tetra chloride molecule, whose abstract group representation is a,b|a4= b3=(ab)2?. [ABSTRACT FROM AUTHOR]

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 (≥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. In this paper, a set of necessary and sufficient conditions for the existence of smooth epimorphism from a Fuchsian group to the point group Oh, which belongs to the sub-class of octahedral group considering the symmetries of Sulfur-Hexafloride (S F6) molecule Having the abstract group representation „'<α² =β4=(αβ)²=1> is established to fulfill the objective. [ABSTRACT FROM AUTHOR]

The symmetry present in molecules is a fundamental concept in Chemistry. Group Theory is an extremely powerful tool which, in spite of abstractness provides the systematic treatment of symmetry of molecules that simplifies the process of obtaining a variety of information about molecules. Molecules are classified according to their symmetry properties. In this paper, analyzing all the symmetry operations as well as symmetry elements of Sulphur-hexa--floride (SF6) molecule, the authors determine the point group and its abstract presentation as <α, β / α² = β4 = (αβ)6 = 1> . [ABSTRACT FROM AUTHOR]