# Papers

Journal of Ultra Scientist of Physical Sciences - A
Voulume: 32
Issue: 3
Article No. 1
Existence of Smooth Epimorphism from Fuchsian Group to the Group of Automorphisms of compact Riemann surface to the point group of Carbon Tetrachloride
MOLOYA BHUYAN
Department of Mathematics, DCB Girlsu2019 College, Jorhat-ASSAM (India)
CHANDRA CHUTIA
Department of Mathematics, Jorhat Institute of Science & Technology, Jorhat-ASSAM
Abstract :

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 $\Gamma$ and an epimorphism $\phi$:$\Gamma$$\rightarrow$G such that ker$\phi$ = K is a surface group of genus g. And then $\phi$ 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 $\Gamma$ to the finite group of symmetries of Carbon Tetra chloride molecule, whose abstract group representation is $<$a,b|a4= b3=(ab)2 $>$.

Keyword : smooth epimorphism, point group, compact Riemann surface, AMS Subject Classification 2020: 20B25, 20B30, 20H10, 30F10