Complete Classification of Degree 7 for Genus 1

Assume that is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r -tuples ( of permutations in the symmetric group , in which and s generate a transitive subgroup G of This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.


INTRODUCTION
A function is a non-constant meromorphic function from compact connected Riemann surface X of genus g to Riemann sphere , if it can locally be described as two holomorphic functions. Furthermore, the mesomorphic function is of degree if the order of the fiber (| ( ) for general equal to . A point is a branch point if (| ( We suppose that is the branch points of the meromorphic function . The fundamental group ( for any acts transitively on n elements of the fiber. This action gives us a homomorphism ( . The image of is called Monodromy group of and denoted by Mon(X, ). We denote for as the closed path winding once around the point . So, the fundamental group ( is generated by the homotopy class of . This generator satisfies the relation , which yields the generators { of monodromy group Mon(X, ), where ( satisfies the following conditions 〈 〉 ( ( is called Riemann Hurwtiz formula where ( on ( . Let G be a transitive subgroup of . A genus g-system is a tuple ( satisfying equations (1), (2) and (3) [1]. A natural question that arises from this study is: what value can monodromy group Mon(X, ) take for a fixed genus of X? Guralnick and Thompson [2], in 1990, gave a conjecture; There is a finite set ( of simple groups, for any positive integer g, such that if is a cover , where X is compact connected Riemann surface of genus g and is a non-commutative composition factor of Mon(X, ), then this composition factor is either an isomorphism to the finite set ( or is

ISSN: 0067-2904
Khudhur Iraqi Journal of Science, 2021, Vol. 62, No. 2, pp: 594-603 595 equal to This fact was established in 2001 by Frohardt and Magaard [1]. As the set ( is finite , one hopes to see the sets ( ( and ( clearly. In 1990, Guralnick and Thompson [2] used Riemann's Existence theorem to prove that the finite set ( takes place in a primitively monodromy action. That is, if G is a group in the set ( then there exists (X, ) in which G= Mon(X, ), and this group acts primitively on the fibers. This result refers to the theorem of Aschbacher and Scott [3]. Theorem1:( Aschbacher and Scott). Let G be a finite group, F(G) be a generated fitting subgroup of the group G, and L be a maximal subgroup of G such that ⋂ Let K be a minimal normal subgroups of G and P be a minimal normal subgroup of K. Let be the set of G conjugates of P. Then G=LK and exactly one of the following holds: 1-P has prime order.
The first case in the above theorem was studied by Guralnick and Thompson [2]. They showed that there are only finitely many primitive affine groups which occur as a composition factor of a primitive genus zero. Furthermore, MohammedSalih [4] studied the same case of genus one and two. Cases 2 and 3 of the above theorem were considered by Shilh [5] and Guralnick and Thompson [2], respectively. They showed that there is no primitive genus zero system. Case 4 of theorem 1 was investigated by Aschbacher [6], who proved that the general fitting subgroup (F(G)) in case of genus zero system must be equal to . MohammedSalih [7] classified the primitive group for genus zero. The final case was studied by Khudhur [8,9,10], who determined all sporadic simple groups of genus zero, one, and two. Our aim in this paper is to determine all primitive genus one groups of degree seven, except . Now we give the following theorem. Theorem 2: Let G denotes a group of degree 7. There exists non-constant meromorphic function from compact connected Riemann surface X of genus one, such that G is a composition factor of Mon(X, ), if and only if ( ( and Next, we will prove theorem 2. In the next lemma, we generalize the above lemma for genus k -system for k>2.

Lemma 2.9: Let G be a finite group and M is a Maximal subgroup of G. Then
If ∑ ∑ ( ( ̂ , then ̂ has not a genus k-system.

3-POSSIBLE RAMIFICATION TYPES
In this section, we find all possible ramification types and use filers to eliminate signature as much as we can. Firstly, we start by using equation 3 to remove the groups ( ( Lemma3.1: Let G be a group ( ( . Then G possesses no primitive genus one system. Proof: The group ( has six conjugacy classes of order 7 but in different types, which are (7A,7B,7C,7D,7E,7F). By using equation 3 It follows that all of them will be eliminated. Next, we present the possible ramification type of the group AGL(1,7). Since the degree operation of AGL(1,7) is equal to seven, then Riemann Hurwitz formula implies that ∑ . such that the product of and is equal to inverse , where x and y are representatives of the conjugacy classes of order three of types A and B, respectively. However, the order of the group generated by the pair ( is not equal to the group AGL (1,7). Hence, the triple (3A,3B,7A) will be ruled out. A GAP calculation shows that the group algebra structure constants [4]  Note that the group ( is of order 168 and it has elements of order 1,2,3,4 and 7 [3]. The group ( acts 2-transitively on seven and eight points. Furthermore, it has two classes of maximal subgroups, which are the symmetric group and 7:3. To determine the list of possible ramification types, we first use the GAP program to find the action of representative of conjugacy classes of ( on the right cosets of the maximal subgroups and 7:3. Lemma 3.6: Let G be a group ( and ̂ , where , acts on the right cosets of the maximal subgroup . Then, and, if then . Proof: Similar to proof 3.4, by using Riemann Hurwitz formula, we obtain the result. By the above lemma, the group ( has 25 possible ramification types when it acts on the right coset of the maximal subgroup . One of these ramification types will be cancelled, which is (2A,7A,7B), because its group algebra structure constant is equal to zero. So, the 24 ramification type cannot be ruled out , as presented in table 1. ) are equal to zero, hence they will be removed. So, 13 ramification types cannot be ignored , which are presented in table 2. The final group with which we work in this paper is the alternating group . The group is of order 2520 and has elements of order 1,2,3,4,5,6 and 7 [3]. It acts 5-transitively on 7 points. The group has four classes of maximal subgroups, which are ( and ( [3]. To determine the list of possible ramification types, similar to the group ( , we first find, using GAP program, the action of representative of conjugacy classes of on the right cosets of the maximal subgroups ( and ( . Lemma3.8: Let G be a group and ̂ , where acts on the right cosets of the maximal subgroup . Then, and, if then . Proof: Similar to proof 3.4, by using Riemann Hurwitz formula, we obtain the result. Lemma3.9: Let G be a group and ̂ , where acts on the right cosets of the maximal subgroup ( and . Then, . Proof: Similar to proof 3.4, by using Riemann Hurwitz formula, we obtain the result. Lemma3.10: Let G be a group and ̂ , where acts on the right cosets of the maximal subgroup ( . . Then, . Proof: Similar to proof 3.4, by using Riemann Hurwitz formula, we obtain the result. By Riemann Hurwitz formula, group when acts on the right cosets of the maximal subgroup has 172 possible ramification types, such that two of them will be removed by using the filter. In  [3]. In this step, we do not study the group because its tuples are too large so that our computers cannot compute the braid orbits. By lemma 3.1, we removed the groups ( ( So, the four groups of ( ( and remained. The first step in this process is to find a list of representative conjugacy classes. In the next step, we give for each group the set of permutation indices. This set determines the tuples which satisfy Riemann Hurwitz formula. The filters presented in section two provided us with the generated ramification types. Next, by using MapClass, we determined the number of components of the Hurwitz spaces, ( . Now, we give the following lemmas. and, if then by Proposition 2.10, G is a cyclic group, hence it will be eliminated. So, all ramification types C are of the length three or four. From Table 2, we observe that all ramification types C up to permutation has only one braid orbit on the Nielsen classes ( ( So, the Hurwitz spaces ( has one component. It follows from the Proposition 2.5 that the Hurwitz spaces ( are connected. Lemma 4.3: For the group L(3,2), the Hurwitz spaces, ( are disconnected if and Proof : By lemma 3.6, the group L(3,2), when it acts on the right coset of the maximal subgroup , has a ramification type of length If then by proposition 2.10, G is cyclic group, and then it will be eliminated. Hence, the length of the ramification types is between . As shown in Table 3, the types C=(3A,4A,7A) and C=(3A,4A,7B) have two braid orbits on Nielsen classes ( ( which implies that the Hurwitz spaces ( has two components. By Proposition 2.5, we obtain that the Hurwitz spaces ( are disconnected. are the disjoint union of braid orbits, but for L (3,2) we have only one braid orbit for all types as given in Table 3. From Proposition 2.5, we obtain that the Hurwitz spaces ( are connected. This means that the length of ramification types is between . Since some type C ramifications in Table 4  Since some type C ramifications in Table 5 have at least two braid orbits on Nielsen classes ( the Hurwitz spaces ( have at least two components. By Proposition 2.5, we obtain that the Hurwitz spaces ( are disconnected. Note that the Hurwitz spaces for all of ramification types C of group are disconnected when acts on the right coset of the maximal subgroup . Lemma 4.7: For the group , the Hurwitz spaces ( are disconnected if and . Proof :By Lemma 3.9, Similar to above, when acts on the right cosets of the maximal subgroup ( has at least two braid orbits for some types on Nielsen classes ( as given in Table 6. Therefore, the Hurwitz spaces ( has at least two components. By proposition 2.5, we obtain that the Hurwitz spaces ( are disconnected.   Tables-(1 to 8).