Genus $g$ Groups of Diagonal Type

Document Type : Research Paper

Author

Department of mathematics, Faculty of science, Soran university, Kawa St., Erbil, Iraq.

Abstract

A transitive subgroup $G\leq S_n$ is called a genus $g$ group if there exist non identity elements $x_1,...,x_r\in G$ satisfying $G=\langle x_1,x_2,...,x_r\rangle$, $\prod_{i=1}^r {x_i}=1$ and $\sum_{i=1}^r ind\, x_i=2(n+g-1)$. The Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ is the space of genus $g$ covers of the Riemann sphere $\mathbb{P}^1\mathbb{C}$ with $r$ branch points and the monodromy group $G$. Isomorphisms of such covers are in one to one correspondence with genus $g$ groups.
In this article, we show that $G$ possesses genus one and two group if it is diagonal type and acts primitively on $\Omega$. Furthermore, we study the connectedness of the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ for genus 1 and 2.

Keywords

References

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Algebraic Structures and Their Applications
Volume 10, Issue 2
August 2023
Pages 41-50

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