## Existence of Smooth Epimorphism from Fuchsian Group to the Group of Automorphisms of compact Riemann surface to the point group of Carbon Tetrachloride

#### AUTHOR AND AFFILIATION

MOLOYA BHUYAN
Department of Mathematics, DCB Girlsu2019 College, Jorhat-ASSAM (India)
CHANDRA CHUTIA
Department of Mathematics, Jorhat Institute of Science & Technology, Jorhat-ASSAM

#### KEYWORDS:

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

March 2020

13-21

#### ISSN:

2319-8044 (Online) – 2231-346X (Print)

Vol.32 – No.3

#### DOI:

http://dx.doi.org/10.22147/jusps-A/320301

#### 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 $>$.