## Complete set of Genera of Compact Riemann Surfaces with reference to the point Group of Sulphur (S8) Molecule

#### AUTHOR AND AFFILIATION

RAFIQUL ISLAM
Department of Mathematics, Jorhat Engineering College, Jorhat-785007 (India)
CHANDRA CHUTIA
Department of Mathematics, Jorhat Institute of Science and Technology, Jorhat (India)

#### KEYWORDS:

Symmetries, Fuchsian group, Smooth quotient, Riemann surface,

July 2020

88-92

#### ISSN:

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

Vol.32 – No.7

#### DOI:

http://dx.doi.org/10.22147/jusps-A/320701

#### ABSTRACT:

In this paper, we consider the group of symmetries of the Sulphur molecule (S) which is a finite point group of order 16 denote by D16 generated by two elements having the presentation { u$\upsilon$/u2$\upsilon$8 = (u$\upsilon$)2 = 1} and find the complete set of genera (g ≥ 2) of Compact Riemann surfaces on which  D16 acts as a group of automorphisms as follows:

D16 the group of symmetries of the sulfur (S8) molecule of order 16 acts as an automorphism group of a compact Riemann surfaces of genus g ≥ 2 if and only if there are integers $\lambda$ and $\mu$ such that $\lambda \leq$ 1 and $\mu \geq$ 1 and
g=$\lambda$ +8$\mu$ ($\geq$2) , $\mu$$\geq$ |$\lambda$|