**Author** : Haval M. Mohammed Salih ^{1&2}

^{1}Soran University, Faculty of Science, Mathematics Department, Soran, Iraq

^{2}Ishik University, Faculty of Education, Mathematics Department, Erbil, Iraq

**Abstract: ** Let be a compact Riemann surface of genus g and µ:X→?^{1} be indecomposable meromorphic function of Riemann sphere by . Isomorphisms of such meromorphic functions are in one to one correspondence with conjugacy classes of r tuples(x_{1 }, x_{2 },…x_{r}) of permutations in S_{n} such that x_{1}, x_{2},…x_{r}=1 and G=< x_{1}, x_{2},…x_{r}> a subgroup of S_{n}. Our goal of this work is to give a classification in the case where X is of genus 1 and the subgroup G is a primitive subgroup of S_{4} or S_{5} . We present the ramification types for genus 1 to complete such a classification. Furthermore, we show that the subgroups D_{10} and C_{5 }of S_{5} do not possesses primitive genus 1 systems.

**Keywords: ** Primitive Groups, Indecomposable Meromorphic Functions, Genus Systems

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**References**

Clebsch, A. (1872). Zur Theorie der Riemann’schen Fläche. 6(2), 216-230.

James, A., Magaard, K., & Shpectorov, S. ( 2012). The lift invariant distinguishes Hurwitz space components for A5. Proceedings of the American Mathematical Society,

Fried, M. (2006) Alternating groups and moduli space lifting invariants. preprint math/0611591.

Gehao, W. (2011). Genus Zero systems for primitive groups of Affine type. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), University of Birmingham.

Liu, F., & Osserman, B. (2008). The irreducibility of certain pure-cylce Hurwitz spaces. Amer. J. Math., 130(6), 1687-1708.

Salih, M. H. (2014) Finite Group of Small Genus. Unpublished Thesis, University of Birmingham.

Salih, M.H., & Akray, I. (2016). Connectedness of the Hurwitz Spaces, A.J. of Garmian University. no 186.

V ̈lklein, H. (1996). Groups as Galois groups an introduction, volume 53 of Cambridge studies in Advanced Mathematics. Cambridge: Cambridge University Press.