SQU Journal for Science, 2017, 22(2), 106-113 DOI: http://dx.doi.org/10.24200/squjs.vol22iss2pp106-113 Sultan Qaboos University 106 Connected Components of 𝓗𝒓,𝒈 𝑨 (𝑮) Haval M. Mohammed Salih Soran University, Faculty of Science, Mathematics Department-Kawa St, Soran, Erbil, Iraq. Email: haval.mahammed@soran.edu.iq ABSTRACT: The Hurwitz space ℋ𝑟,𝑔 𝐴 (𝐺) is the space of genus g covers of the Riemann sphere ℙ1 with 𝑟 branch points and the monodromy group 𝐺. In this paper, we enumerate the connected components of the Hurwitz spaces ℋ𝑟,𝑔 𝐴 (𝐺) for a finite primitive group 𝐺 of degree 7 and genus zero except 𝑆7. We achieve this with the aid of the computer algebra system GAP and the MAPCLASS package. Keywords: Monodromy Groups; Braid orbits; Connected Components. 𝓗𝒓,𝒈 𝑨 (𝑮) المكونات المتصلة ل محمد صالح م. افاله ℋr,gالفضاء الهوروتيزي :صالملخ A (G) هي نوع من غالف في الفضاء الريمانيℙ1 الزمرة المونودروميةمع نقاط تفرع و G. نحن في هذا البحث ℋr,gعددنا المكونات المترابطة لمجموعات البدائية ل A (G) عداوالجنس صفرى من الدرجة .S7 و أحرز هذا الغرض من خالل النظام الحاسوبي MAPCLASSو حزمة GAPالجبري المكونات المتصلة.و مدارات بريد، المنودروميةة زمر :مفتاحيةالكلمات ال 1. Introduction et Ω be a finite set and |Ω| = 𝑛. Define a genus 𝑔 system to be a triple (𝐺, Ω, (𝑥1 , … , 𝑥𝑟 )). 𝐺 is a transitive subgroup of 𝑆𝑛 such that 𝐺 =< 𝑥1, … , 𝑥𝑟 >, 𝑥1 ∙ 𝑥2 ∙ … ∙ 𝑥𝑟 = 1 and 𝑥𝑖 ∈ 𝐺\{1} 2(𝑛 + 𝑔 − 1) = ∑ 𝑖𝑛𝑑𝑥𝑖 𝑟 𝑖=1 where indxi is the minimal number of transpositions need to express 𝑥𝑖 as a product [6]. This condition is equivalent to the existence of the branched covering 𝑓: 𝑋 → ℙ1 where ℙ1 = ℂ ∪ {∞} [8]. If 𝑓 is an irreducible, then 𝐺 is primitive. Let 𝐶1, … , 𝐶𝑟 be non-trivial conjugacy classes of a finite group 𝐺. The set of generating systems (𝑥1, … , 𝑥𝑟 ) of 𝐺 with 𝑥1 … 𝑥𝑟 = 1 and such that there is a permutation 𝜋 ∈ 𝑆𝑟 with 𝑥𝑖 ∈ 𝑆𝜋(𝑖) for 𝑖 = 1, … , 𝑟 is called a Nielsen class and denoted by 𝒩(𝐶), where 𝐶 = (𝐶1, … , 𝐶𝑟 ). Each Nielsen class is the disjoint union of braid orbits, which are defined as the smallest subsets of the Nielsen class closed under the braid operations [10] (𝑥1, … , 𝑥𝑟 ) 𝑄𝑖 = (𝑥1, … , 𝑥𝑖+1, 𝑥𝑖+1 −1 𝑥𝑖 𝑥𝑖+1, … , 𝑥𝑟 ) (1) for 𝑖 = 1, … , 𝑟. We denote by 𝑂𝑟 , the space of subsets of ℂ of cardinality 𝑟. The following definitions can be found in [10]. L mailto:haval.mahammed@soran.edu.iq CONNECTED COMPONENTS OF ℋ𝑟,𝑔 𝐴 (𝐺) 107 Definition 1.1 Let 𝐵 ∈ 𝑂𝑟 and 𝑏0 ∈ ℙ 1 ∖ 𝐵, we call a map 𝜑: 𝜋1(ℙ 1 ∖ 𝐵, 𝑏0) → 𝐺 admissible if it is a surjective homomorphsim, and 𝜑(𝜃𝑏 ) ≠ 1 for each 𝑏 ∈ 𝐵. Here 𝜃𝑏 is the conjugacy class of 𝜋1(ℙ 1 ∖ 𝐵, 𝑏0). Definition 1.2. Let 𝐵 ∈ 𝑂𝑟 and 𝜑: 𝜋1(ℙ 1 ∖ 𝐵, ∞) → 𝐺 be admissible. Then we say that two pairs (𝐵, 𝜑) and (B̅, φ̅) are A- equivalent if and only if B = B̅ and φ̅ = a ∘ φ for some a ∈ A. Let [𝐵, 𝜑]𝐴 denote the 𝐴-equivalence class of (𝐵, 𝜑). The set of equivalence classes [𝐵, 𝜑]𝐴 is denoted by ℋ𝑟 𝐴(𝐺) and is called the Hurwitz space of 𝐺-covers. Here we enumerate the connected components of ℋ𝑟 𝐴 (𝐺) and then we show to which number of branch points 𝑟, it is connected. The MAPCLASS package of James, Magaard, Shpectorov and Volklein, is designed to perform braid orbit computations for a given finite group and given type. 2. Preliminary As usual 𝐼𝑛𝑛(𝐺) and 𝐴𝑢𝑡(𝐺) denote the inner-automorphism and automorphism groups of a group 𝐺 respectively. A denotes a subgroup of 𝐴𝑢𝑡(𝐺). In particular if 𝐴 = 𝐼𝑛𝑛(𝐺), then the Hurwitz space ℋ𝑟 𝐴 (𝐺) is denoted by ℋ𝑟 𝑖𝑛 (𝐺).The details of the following results and concepts can be found in [10] and [8]. Lemma 2.1. The map Ψ𝐴 : ℋ𝑟 𝐴 (𝐺) → 𝑂𝑟, Ψ𝐴([𝐵, 𝜑]) = 𝐵 is covering. The fiber ΨA −1(B0) = {[B0, φ]A: φ: π1(ℙ 1 ∖ B, ∞) → 𝐺 is admissible}. This φ gives a product one generating tuple (x1, … , xr) of G. Define ℰr(G) = {(𝑥1, … , 𝑥𝑟 ): 𝐺 = 〈𝑥1, … , 𝑥𝑟 〉, 𝑥1 ∙ … ∙ 𝑥𝑟 = 1, 𝑥𝑖 ∈ 𝐺 #, 𝑖 = 1,2, … , 𝑟}. Let A ≤ Aut(G). Then the subgroup A acts on ℰr(G) via sending (x1, … , xr) to (a(x1), … , a(xr)), for a ∈ A. Let ℰ𝑟 𝐴 (𝐺) = ℰ𝑟 (𝐺)/𝐴 be the set of 𝐴-orbits. In particular, if 𝐴 = 𝐼𝑛𝑛(𝐺), then we have 𝐼𝑛𝑛(𝐺) = 𝐺/𝑍(𝐺). Therefore ℰ𝑟 𝑖𝑛 (𝐺) is the set of 𝐺-orbits. Lemma 2.2. We obtain a bijection Ψ𝐴 −1(𝐵0) → ℰ𝑟 𝐴 (𝐺) by sending [𝐵0, 𝜑]𝐴 to the generators (𝑥1, … , 𝑥𝑟 ) where 𝑥𝑖 = 𝜑([𝛾𝑖 ]) for 𝑖 = 1, … , 𝑟. Proposition 2.3. Let C be a fixed ramifcation type in G, and the subset ℋr in(C) of ℋr in(G) consists of all [B, ∅]A with B = {b1, … , br}, ∅: π1(ℙ 1\B, ∞) → G and ∅(θbi ) ∈ Ci for i = 1, … , r. Then ℋr A(C) is a union of connected components in ℋr A(G). Under the bijection from Lemma 2.2, the fiber in ℋr A(C) over B0 corresponds the set 𝒩 A (C). This yields a one to one correspondence between components of ℋr A (C) and the braid orbits on 𝒩 A (C). In particular, ℋr in(C) is connected if and only if Bracts transitively on 𝒩 in(C) = 𝒩(C) . Lemma 2.4. Let 𝐺 be a group and 𝑋 be a 𝐺-set. Then 𝐺 acts transitively on 𝑋 if and only if there is only one orbit. Proof. Straightforward. Corollary 2.5. Let C be a fixed ramifcation type in 𝐺, and the subset ℋr in(C) of ℋr in(G) consists of all [B, ∅]A with B = {b1, … , br}, ∅: π1(ℙ 1\B, ∞) → 𝐺 and ∅(θbi ) ∈ Ci for i = 1, … , r. Then ℋr A(C) is a union of connected components in ℋr A(𝐺). Under the bijection from Lemma 2.2, the fiber in ℋr A(C) over B0 corresponds the set 𝒩 A (C). This yields a one to one correspondence between components of ℋr A (C) and the braid orbits on 𝒩 A (C). In particular, ℋr in(C) is connected if and only if there is only one braid orbit. Proof. It follows from Proposition 2.3 and Lemma 2.4. 3. Computing Indices and Labeling Conjugacy Classes The classifications of all the primitive groups of degree 7 except S7 for genus zero are given in this paper. Before, we discuss computing the indices, we give an alternative formula to compute an index of an element in a group. Let G be a group acting on a finite set Ω and |Ω| = n. If 𝑥 ∈ 𝐺, define the index of 𝑥 by 𝑖𝑛𝑑 𝑥 = 𝑛 − 𝑜𝑟𝑏 𝑥, where orb x is the number of orbits of < 𝑥 > on Ω. Also Fix 𝑥={𝑤 ∈ 𝛺 | 𝑥𝑤 = 𝑤}, 𝑓(𝑥) = |𝐹𝑖𝑥 𝑥|. Furthermore, 𝑜𝑟𝑏 𝑥 = 1 𝑑 ∑ 𝑓(𝑥 𝑖 )𝑑−1𝑖=0 where 𝑥 has order 𝑑 [6]. From the character table of A7, we see the elements of orders 2,3,4,5,6 and 7, then we compute fixed points, which are equal to 1a+2a of the elements of given orders. HAVAL M. MOHAMMED SALIH 108 The Character Table of 𝐴7. Alternating group 𝐴7 Order =2520=23. 32. 5.7 mult =6 out = 2 Constructions Alternating 𝑆7 ≅ 𝐺. 2 ∶ all permutations of 7 letters; 𝐴7 ≅ 𝐺 ∶ the even permutations; 2. 𝐺 and 2. 𝐺. 2 ∶ the schur double covers Lattice 2𝐴7 ≅ 2. 𝐺 ∶ the symmetries of the lattice ⋀4,𝑏7 whose minimal vectors are obtained from ±(𝑖7; 0,0,0) ± (0; 𝑖7,0,0) ± (1; , 𝑥, 𝑦, 𝑧) ± (−𝑥; 1, 𝑦, −𝑧) ± (−𝑥𝑦; ,1,1,1) ± (1; 𝑥𝑦, 1,1) ± (𝑥𝑦; 0, 𝑧, 1) ±(−𝑥; 𝑦𝑧, 0,1) ± (−1; 0, 𝑥𝑦, 𝑧) ± (0; 1, −𝑥, 𝑦𝑧) by replacing each of 𝑥, 𝑦, 𝑧 by one of 𝑏7 or 𝑏7 ∗∗, and Cyclically permuting the last 3 coordinates Vectors 3𝐴7 ∶ symmetries of the 21 (𝑤) vectors obtained from (200 00 00), (00 11 11), (01 01 𝑤�̅�) by Bodiy permuting the 3 couples, and reversing any 2 couples (see 𝐴6 (hexacode)); The lattice these generate has auomorphism group 6𝑈4(3). 2 (see 𝑈4(3)) Unitary 3𝐴7 has a 3-dimensional unitary representation over 𝐹25 (see 𝑈3(5)) Presentations 𝐺 ≅< 𝐴, 𝐵| 𝐴4 = 𝐵5 = (𝐴𝐵)3 = (𝐴−1𝐵𝐴2𝐵2)2 = 1 >≅< 𝑥1, … , 𝑥5|𝑥𝑖 3 = (𝑥𝑖 𝑥𝑗 ) 2 = 1 >; 𝐺 ≅< 𝐴, 𝐵| 𝐴2 = [𝐴, 𝐵2]2 = [𝐴, 𝐵3]2 = 1; 𝐵7 = (𝐴𝐵)6 > Maximal subgroups specifications Order Index Structure 𝐺. 2 Character Abstract Alternating 360 7 𝐴6 : 𝑆6 1𝑎 + 2𝑎 𝑁(2𝐴, 3𝐴, 3𝐵, 4𝐴, 5𝐴) point 168 15 𝐿2(7) 7: 6 1𝑎 + 14𝑏 𝑁(2𝐴, 3𝐵, 4𝐴, 7𝐴𝐵) 𝑆(2,3,7) 168 15 𝐿2(7) 7: 6 1𝑎 + 14𝑏 𝑁(2𝐴, 3𝐵, 4𝐴, 7𝐴𝐵) 𝑆(2,3,7) 120 21 𝑆5 : 𝑆5 × 2 1𝑎 + 6𝑎 + 14𝑎 𝑁(2𝐴, 3𝐴, 5𝐴), 𝐶(2𝐵) dual 72 35 (𝐴4 × 3): 2 : 𝑆4 × 𝑆3 1𝑎 + 6𝑎 + 14𝑎𝑏 𝑁(3𝐴), 𝑁(2𝐴 2) triad CONNECTED COMPONENTS OF ℋ𝑟,𝑔 𝐴 (𝐺) 109 ; @ @ @ @ @ @ @ @ @ : : @ @ @ @ @ @ @ 𝑝 𝑝′ ind 2520 power part 1A 24 A A 2A 36 A A 3A 9 A A 3B 4 A A 4A 5 A A 5A 12 AA AA 6A 7 A A 7A 7 A A 𝐵∗∗ fus ind 120 A A 2B 24 A A 2C 12 A A 4B 6 AB AB 6B 3 BC BC 6C 5 AB AB 10A 6 AB AB 12A 𝒳1 + 1 1 1 1 1 1 1 1 1 : ++ 1 1 1 1 1 1 1 𝒳2 + 6 2 3 0 0 1 -1 -1 -1 : ++ 4 0 2 1 0 -1 -1 𝒳3 O 10 -2 1 1 0 0 1 𝑏7 ** + 0 0 0 0 0 0 0 𝒳4 O 10 -2 1 1 0 0 1 ** 𝑏7 𝒳5 + 14 2 2 -1 0 -1 2 0 0 : ++ 6 2 0 0 -1 1 0 𝒳6 + 14 2 -1 2 0 -1 -1 0 0 : ++ 4 0 -2 1 0 -1 1 𝒳7 + 15 -1 3 0 -1 0 -1 1 1 : ++ 5 -3 1 -1 0 0 1 𝒳8 + 21 1 -3 0 -1 1 1 0 0 : ++ 1 -3 -1 1 0 1 -1 𝒳9 + 35 -1 -1 -1 1 0 -1 0 0 : ++ 5 1 -1 -1 1 0 -1 If 𝑥 is an element of order 2, then 𝑖𝑛𝑑 𝑥 = 𝑛 − 1 2 ∑ 𝑓(𝑥 𝑖 ) = 𝑛 − 1 2 [𝑓(𝑥 0) + 𝑓(𝑥)] = 7 − 1 2 [7 + 3] = 21𝑖=0 . If 𝑥 is an element of order 3 of type 3𝐴, then 𝑖𝑛𝑑 𝑥 = 𝑛 − 1 3 ∑ 𝑓(𝑥 𝑖) = 𝑛 − 1 3 [𝑓(𝑥 0) + 𝑓(𝑥) + 𝑓(𝑥 2)] = 7 − 1 3 [7 + 4 + 4] = 22𝑖=0 . If 𝑥 is an element of order 3 of type 3𝐵, then 𝑖𝑛𝑑 𝑥 = 𝑛 − 1 3 ∑ 𝑓(𝑥 𝑖) = 𝑛 − 1 3 [𝑓(𝑥 0) + 𝑓(𝑥) + 𝑓(𝑥 2)] = 7 − 1 3 [7 + 1 + 1] = 42𝑖=0 . If 𝑥 is an element of order 4, then 𝑖𝑛𝑑 𝑥 = 𝑛 − 1 4 ∑ 𝑓(𝑥 𝑖 ) = 𝑛 − 1 4 [𝑓(𝑥 0) + 𝑓(𝑥) + 𝑓(𝑥 2) + 𝑓(𝑥 3)] = 7 − 1 4 [7 + 1 + 3 + 1] = 43𝑖=0 . If 𝑥 is an element of order 5, then 𝑖𝑛𝑑 𝑥 = 𝑛 − 1 5 ∑ 𝑓(𝑥 𝑖 ) = 𝑛 − 1 5 [𝑓(𝑥 0) + 𝑓(𝑥) + 𝑓(𝑥 2) + 𝑓(𝑥 4)] = 7 − 1 5 [7 + 2 + 2 + 2 + 2] = 44𝑖=0 . If 𝑥 is an element of order 6, then 𝑖𝑛𝑑 𝑥 = 𝑛 − 1 6 ∑ 𝑓(𝑥 𝑖) = 𝑛 − 1 6 [𝑓(𝑥 0) + 𝑓(𝑥) + 𝑓(𝑥 2) + 𝑓(𝑥 4) + 𝑓(𝑥 5)] = 7 − 1 6 [7 + 0 + 4 + 3 + 4 + 0] = 45𝑖=0 . If 𝑥 is an element of order 7 of type 7𝐴 or 7𝐵, then 𝑖𝑛𝑑 𝑥 = 𝑛 − 1 7 ∑ 𝑓(𝑥 𝑖 ) = 𝑛 − 1 7 [𝑓(𝑥 0) + 𝑓(𝑥) + 𝑓(𝑥 2) + 𝑓(𝑥 4) + 𝑓(𝑥 5) + 𝑓(𝑥 6)] = 7 − 1 7 [7 + 0 + 0 + 0 + 0 +6𝑖=0 0 + 0] = 6. 4. Algorithm and Main Results To obtain Tables 2 and 3, we need to perform the following steps: 1- We extract all primitive permutation groups 𝐺 by using the GAP function [4] AllPrimitiveGroups (DegreeOperation,𝑛). 2- For given degree, genus and 𝐺 we compute all possible ramification types satisfying the Riemann-Hurwitz formula which is given in section 3. 3- We compute the character table of 𝐺 and remove those types which have zero structure constant. 4- We obtain all generating types by GAP Codes which exist in Appendix C [8]. 5- For each of the remaining generating types of length greater than or equal to 4, we use the MAPCLASS package to compute braid orbits. For tuples of length 3, we determine braid orbits via double cosets in [8]. We now give our main results as follows: Lemma 4.1. The Hurwitz spaces, ℋ𝑟 𝑖𝑛 (𝐶) are connected if 𝐺 = 𝐷(2 ∗ 7) or 𝐺 = 𝐴𝐺𝐿(1,7). Proof. It follows from the fact that the Nielsen classes 𝒩(𝐶) are the disjoint union of braid orbits but we have only one braid orbit for 𝑟 ≥ 3 and 𝑛 = 7. From Corollary 2.5, we obtain the Hurwitz spaces ℋ𝑟 𝑖𝑛 (𝐶) are connected. Lemma 4.2. The Hurwitz spaces, ℋ𝑟 𝑖𝑛 (𝐶) are connected if 𝑟 ≥ 4 and 𝐺 = 𝐿(3,2). Proof. It follows from the fact that the Nielsen classes 𝒩(𝐶) are the disjoint union of braid orbits but we have only one braid orbit for 𝑟 ≥ 4 and 𝑛 = 7. From Corollary 2.5, we obtain the Hurwitz spaces ℋ𝑟 𝑖𝑛 (𝐶) are connected. Lemma 4.3. The Hurwitz spaces, ℋ𝑟 𝑖𝑛 (𝐶) are disconnected if 𝐺 = 𝐴7 and 𝐺 = 𝐶7. Proof. It follows from the fact that the Nielsen classes 𝒩(𝐶) are the disjoint union of braid orbits but for these groups we have at least two braid orbits for some type 𝐶 as given in Table 2. From Corollary 2.5, we obtain the Hurwitz spaces ℋ𝑟 𝑖𝑛 (𝐶) are disconnected. HAVAL M. MOHAMMED SALIH 110 Finally, we enumerate the connected components of ℋ𝑟 𝑖𝑛 (𝐺) in the cases where 𝑔 = 0 and 𝐺 is a primitive group of degree 7. The total numbers of connected components of ℋ𝑟 𝑖𝑛 (𝐺) is summarized in Table 1. Table 1. Primitive Genus Zero Systems: Number of Components. Degree # Group Iso types #RTs # comp’s r=3 # comp’s r=4 # comp’s r=5 # comp’s r=6 # comp’s Total 7 5 154 179 61 67 10 317 Example 4.4 First of all, read the file in GAP program which exists in [8] and then choose the group and the specific tuple. For instance let 𝐺 = 𝐷(2 ∗ 7) be the dihedral group and take the tuple 𝑡 = [(2,5)(3,6)(4,7), (2,5)(3,6)(4,7), (2,5)(3,6)(4,7), (2,5)(3,6)(4,7)] . The run of the program which finds the braid orbits is shown below: gap> Read("qu1.g"); ---------------------------------------------------------------- Loading MapClass 1.2 by Adam James (http://www.mat.bham.ac.uk/~jamesa) Kay Magaard (http://mat.bham.ac.uk/staff/magaardk.shtml) Sergey Shpectorov (http://web.mat.bham.ac.uk/S.Shpectorov/index.html) Helmut Volklein (http://www.iem.uni-due.de/algebra/people/voelklein.html) For help, type: ?MapClass: ---------------------------------------------------------------- gap> LL:=AllPrimitiveGroups(DegreeOperation,7); [ C(7), D(2*7), 7:3, AGL(1, 7), L(3, 2), A(7), S(7) ] gap> k:=LL[2]; D(2*7) gap> CheckingTheGroup(k); gap> GT:=GeneratingType(k,7,0); Checking the ramification type 10 with 0 remaining [ [ 2, 2, 5 ], [ 2, 2, 4 ], [ 2, 2, 3 ], [ 2, 2, 2, 2 ] ] gap> t:=List(GT[4],x->CC[x]); [ (2,5)(3,6)(4,7), (2,5)(3,6)(4,7), (2,5)(3,6)(4,7), (2,5)(3,6)(4,7) ] gap> orb:=GeneratingMCOrbits(k,0,t);; The current date is: Mon 07/10/2017 Enter the new date: (mm-dd-yy) Total Number of Tuples: 336 Collecting 20 generating tuples .. done Cleaning done; 20 random tuples remaining Orbit1: Length=24 Generating Tuple =[ (1,4)(3,5)(6,7), (1,6)(2,3)(5,7), (1,7)(2,6)(3,4), (1,3)(2,4)(5,6) ] Centralizer size=1 0 tuples remaining Cleaning a list of 20 tuples Random Tuples Remaining: 0 Cleaning done; 0 random tuples remaining Computation complete: 1 orbits found. http://mat.bham.ac.uk/staff/magaardk.shtml CONNECTED COMPONENTS OF ℋ𝑟,𝑔 𝐴 (𝐺) 111 Table 2. Primitive Groups of Degree 7. Group Ramification Type # of orbits Length of largest orbit Ramification Type # of orbits Length of largest orbit 𝐴7 (5𝐴, 5𝐴, 5𝐴) 2 1 (4𝐴, 5𝐴, 5𝐴) 8 1 (4𝐴, 4𝐴, 5𝐴) 22 1 (4𝐴, 4𝐴, 4𝐴) 24 1 (3𝐵, 5𝐴, 5𝐴) 6 1 (3𝐵, 4𝐴, 5𝐴) 10 1 (3𝐵, 4𝐴, 4𝐴) 8 1 (3𝐵, 3𝐵, 5𝐴) 2 1 (5𝐴, 5𝐴, 6𝐴) 8 1 (4𝐴, 5𝐴, 6𝐴) 8 1 (4𝐴, 4𝐴, 6𝐴) 12 1 (3𝐵, 5𝐴, 6𝐴) 6 1 (3𝐵, 4𝐴, 6𝐴) 6 1 (3𝐵, 3𝐵, 6𝐴) 2 1 (6𝐴, 6𝐴, 6𝐴) 2 1 (3𝐵, 6𝐴, 6𝐴) 2 1 (4𝐴, 6𝐴, 6𝐴) 4 1 (5𝐴, 6𝐴, 6𝐴) 2 1 (3𝐴, 5𝐴, 7𝐴) 1 1 (3𝐴, 5𝐴, 7𝐵) 1 1 (3𝐴, 4𝐴, 7𝐴) 2 1 (3𝐴, 5𝐴, 7𝐵) 2 1 (3𝐴, 3𝐵, 7𝐴) 1 1 (3𝐴, 3𝐵, 7𝐵) 1 1 (3𝐴, 6𝐴, 7𝐴) 1 1 (3𝐴, 6𝐴, 7𝐵) 1 1 (2𝐴, 5𝐴, 7𝐴) 2 1 (2𝐴, 5𝐴, 7𝐵) 2 1 (2𝐴, 4𝐴, 7𝐴) 2 1 (2𝐴, 4𝐴, 7𝐵) 2 1 (2𝐴, 6𝐴, 7𝐴) 2 1 (2𝐴, 6𝐴, 7𝐵) 2 1 (3𝐴, 3𝐴, 5𝐴, 5𝐴) 1 30 (3𝐴, 3𝐴, 4𝐴, 5𝐴) 2 40 (3𝐴, 3𝐴, 4𝐴, 4𝐴) 2 92 (3𝐴, 3𝐴, 3𝐵, 5𝐴) 1 40 (3𝐴, 3𝐴, 3𝐵, 4𝐴) 2 44 (3𝐴, 3𝐴, 3𝐵, 3𝐵) 1 26 (3𝐴, 3𝐴, 6𝐴, 5𝐴) 1 60 (3𝐴, 3𝐴, 6𝐴, 4𝐴) 1 72 (3𝐴, 3𝐴, 3𝐵, 6𝐴) 1 40 (3𝐴, 3𝐴, 6𝐴, 6𝐴) 1 22 (3𝐴, 3𝐴, 3𝐴, 7𝐴) 1 7 (3𝐴, 3𝐴, 3𝐴, 7𝐵) 1 7 (2𝐴, 3𝐴, 5𝐴, 5𝐴) 1 80 (2𝐴, 3𝐴, 4𝐴, 5𝐴) 1 170 (2𝐴, 3𝐴, 4𝐴, 4𝐴) 1 300 (2𝐴, 3𝐴, 3𝐵, 5𝐴) 1 90 (2𝐴, 3𝐴, 3𝐵, 4𝐴) 1 126 (2𝐴, 3𝐴, 3𝐵, 3𝐵) 1 62 (2𝐴, 3𝐴, 6𝐴, 5𝐴) 1 80 (2𝐴, 3𝐴, 4𝐴, 6𝐴) 1 118 (2𝐴, 3𝐴, 3𝐵, 6𝐴) 1 62 (2𝐴, 3𝐴, 6𝐴, 6𝐴) 1 44 (2𝐴, 3𝐴, 3𝐴, 7𝐴) 1 14 (2𝐴, 3𝐴, 6𝐴, 7𝐵) 1 14 (2𝐴, 2𝐴, 5𝐴, 5𝐴) 3 70 (2𝐴, 2𝐴, 4𝐴, 5𝐴) 3 120 (2𝐴, 2𝐴, 4𝐴, 4𝐴) 3 168 (2𝐴, 2𝐴, 3𝐵, 5𝐴) 1 150 (2𝐴, 2𝐴, 3𝐵, 4𝐴) 1 192 (2𝐴, 2𝐴, 3𝐵, 3𝐵) 1 44 (2𝐴, 2𝐴, 6𝐴, 5𝐴) 3 60 (2𝐴, 2𝐴, 6𝐴, 4𝐴) 3 76 (2𝐴, 2𝐴, 3𝐵, 6𝐴) 1 90 (2𝐴, 2𝐴, 6𝐴, 6𝐴) 3 36 (2𝐴, 2𝐴, 3𝐴, 7𝐴) 1 28 (2𝐴, 2𝐴, 3𝐴, 7𝐵) 1 28 (2𝐴, 2𝐴, 2𝐴, 7𝐴) 2 21 (2𝐴, 2𝐴, 2𝐴, 7𝐵) 2 21 (2𝐴, 2𝐶, 3𝐴, 3𝐴, 4𝐴) 1 168 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 6𝐴) 1 300 (2𝐴, 2𝐶, 3𝐴, 3𝐴, 2𝐵) 1 42 (2𝐴, 3𝐴, 3𝐴, 3𝐴, 4𝐴) 1 96 (2𝐴, 3𝐴, 3𝐴, 3𝐴, 6𝐴) 1 216 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 2𝐵) 1 44 (2𝐴, 2𝐶, 2𝐶, 3𝐴, 4𝐴) 1 240 (2𝐴, 2𝐶, 2𝐶, 3𝐴, 6𝐴) 1 384 (2𝐴, 2𝐶, 2𝐶, 2𝐶, 3𝐴) 1 57 (2𝐴, 2𝐶, 2𝐶, 2𝐶, 4𝐴) 1 312 (2𝐴, 2𝐶, 2𝐶, 2𝐶, 6𝐴) 1 486 (2𝐴, 2𝐶, 2𝐶, 2𝐶, 2𝐵) 1 60 (2𝐴, 2𝐴, 3𝐴, 4𝐴, 4𝐴) 1 89 (2𝐴, 2𝐴, 3𝐴, 4𝐴, 6𝐴) 1 202 (2𝐴, 2𝐴, 3𝐴, 6𝐴, 6𝐴) 1 336 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 5𝐴) 1 75 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 4𝐴) 1 80 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 3𝐵) 1 39 (2𝐴, 2𝐴, 2𝐵, 3𝐴, 4𝐴) 1 36 (2𝐴, 2𝐴, 2𝐵, 3𝐴, 6𝐴) 1 52 (2𝐴, 2𝐴, 2𝐶, 4𝐴, 4𝐴) 1 158 (2𝐴, 2𝐴, 2𝐶, 4𝐴, 6𝐴) 1 273 (2𝐴, 2𝐴, 2𝐶, 6𝐴, 6𝐴) 1 426 (2𝐴, 2𝐴, 2𝐶, 3𝐴, 5𝐴) 1 125 (2𝐴, 2𝐴, 2𝐶, 3𝐴, 4𝐴) 1 100 (2𝐴, 2𝐴, 2𝐶, 3𝐴, 3𝐵) 1 48 (2𝐴, 2𝐴, 2𝐶, 2𝐵, 4𝐴) 1 40 (2𝐴, 2𝐴, 2𝐵, 2𝐶, 6𝐴) 1 60 HAVAL M. MOHAMMED SALIH 112 (2𝐴, 2𝐴, 2𝐶, 2𝐶, 5𝐴) 1 175 (2𝐴, 2𝐴, 2𝐶, 2𝐶, 4𝐴) 1 128 (2𝐴, 2𝐴, 2𝐶, 2𝐶, 3𝐵) 1 54 (2𝐴, 2𝐴, 2𝐴, 4𝐴, 5𝐴) 1 75 (2𝐴, 2𝐴, 2𝐴, 4𝐴, 5𝐴) 1 72 (2𝐴, 2𝐴, 2𝐴, 4𝐴, 4𝐴) 1 36 (3𝐴, 3𝐴, 3𝐴, 3𝐴, 5𝐴) 1 300 (3𝐴, 3𝐴, 3𝐴, 3𝐴, 4𝐴) 2 384 (3𝐴, 3𝐴, 3𝐴, 3𝐴, 3𝐵) 1 312 (3𝐴, 3𝐴, 3𝐴, 3𝐴, 6𝐴) 1 480 (2𝐴, 3𝐴, 3𝐴, 3𝐴, 5𝐴) 1 750 (2𝐴, 3𝐴, 3𝐴, 3𝐴, 4𝐴) 1 1392 (2𝐴, 3𝐴, 3𝐴, 3𝐴, 3𝐵) 1 744 (2𝐴, 3𝐴, 3𝐴, 3𝐴, 6𝐴) 1 690 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 5𝐴) 1 1550 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 4𝐴) 1 2704 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 3𝐵) 1 1234 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 6𝐴) 1 1112 (2𝐴, 2𝐴, 2𝐴, 3𝐴, 5𝐴) 1 3000 (2𝐴, 2𝐴, 2𝐴, 3𝐴, 4𝐴) 1 4584 (2𝐴, 2𝐴, 2𝐴, 3𝐴, 3𝐵) 1 2214 (2𝐴, 2𝐴, 2𝐴, 3𝐴, 5𝐴) 1 1896 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 5𝐴) 3 1800 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 4𝐴) 3 2880 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 3𝐵) 1 3240 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 6𝐴) 3 1080 (2𝐴, 2𝐴, 3𝐴, 3𝐴, 3𝐴, 3𝐴) 1 13764 (2𝐴, 3𝐴, 3𝐴, 3𝐴, 3𝐴, 3𝐴) 1 7280 (3𝐴, 3𝐴, 3𝐴, 3𝐴, 3𝐴, 3𝐴) 1 2870 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 3𝐴, 3𝐴) 1 45692 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 2𝐴, 3𝐴) 1 79560 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 2𝐴, 2𝐴) 3 45360 (2𝐴, 2𝐴, 2𝐴, 3𝐴, 3𝐴, 3𝐴) 1 26210 Table 3. Primitive Groups of Degree 7. Group Ramification Type # of orbits Length of largest orbit Ramification Type # of orbits Length of largest orbit 𝐴𝐺𝐿(1,7) (2𝐴, 3𝐵, 6𝐵) 1 1 (2𝐴, 3𝐴, 6𝐴) 1 1 𝐿(3,2) (3𝐴, 3𝐴, 4𝐴) 4 1 (3𝐴, 4𝐴, 4𝐴) 2 1 (4𝐴, 4𝐴, 4𝐴) 4 1 (2𝐴, 3𝐴, 7𝐵) 1 1 (2𝐴, 3𝐴, 7𝐴) 1 1 (2𝐴, 4𝐴, 7𝐵) 1 1 (2𝐴, 4𝐴, 7𝐴) 1 1 (2𝐴, 2𝐴, 3𝐴, 3𝐴) 1 30 (2𝐴, 2𝐴, 3𝐴, 4𝐴) 1 24 (2𝐴, 2𝐴, 4𝐴, 4𝐴) 1 24 (2𝐴, 2𝐴, 2𝐴, 7𝐴) 1 1 (2𝐴, 2𝐴, 2𝐴, 7𝐵) 1 7 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 3𝐴) 1 216 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 4𝐴) 1 192 (2𝐴, 2𝐴, 2𝐴, 2𝐴, 2𝐴) 1 1680 𝐷(2 ∗ 7) (2𝐴, 2𝐴, 7𝐴) 1 1 (2𝐴, 2𝐴, 7𝐵) 1 1 (2𝐴, 2𝐴, 7𝐶) 1 1 (2𝐴, 2𝐴, 2𝐴, 2𝐴) 1 24 𝐶7: 𝐶3 (3𝐵, 3𝐵, 3𝐵) 2 1 (3𝐴, 3𝐴, 3𝐴) 2 1 5. Conclusion In this paper, we use the algorithm in [8] to compute braid orbits on Nielsen class. An application of the algorithm is the classification of the primitive genus zero systems of degree 7. That is we find the connected components ℋ𝑟 𝑖𝑛 (𝐺) of 𝐺-curves 𝑋, such that 𝑔 = 0. In our situation, the computation shows that there are exactly 307 braid orbits of primitive genus 0 systems of degree 7. References 1. Michael, D.F. and Helmut, V. The inverse Galois problem and rational points on moduli spaces. Mathematische Annalen, 1991, 290(4), 771-800. 2. Daniel, F., Robert, G. and Kay, M. Genus 0 actions of groups of Lie rank 1. American Mathematical Society, Providence, Rhode Island, Proceedings of Symposia in Pure Mathematics, 2002,70, 449-483. 3. Daniel, F. and Kay, M. Composition factors of monodromy groups. Annals of Mathematics. Second Series. 2001, 154(2), 327-345. 4. The GAP Group. GAP-Groups, Algorithms, and Programming, Version 4.6.2, 2013. http://www.gap-system.org 5. Wang, G. Genus Zero Systems for Primitive Groups of Affine Type. 2011. Ph.D Thesis University of Birmingham, UK. 6. Robert, M.G. and John, G.T. Finite groups of genus zero. Journal of Algebra, 1990, 131(1), 303-341. 7. Magaard, K., Shpectorov, S. and Wang, G. Generating sets of affine groups of low genus. In Computational algebraic and analytic geometry, American Mathematical Society, Providence, Rhode Island, 2012, 572, 173-192. CONNECTED COMPONENTS OF ℋ𝑟,𝑔 𝐴 (𝐺) 113 8. Haval, M.S. Finite Groups of Small Genus. 2014. Ph.D Thesis University of Birmingham, UK. 9. Michael, G.N. On solvable monodromy groups of fixed genus. Pro-Quest LLC, Ann Arbor, Michigan, 1989. Ph.D Thesis University of Southern California, USA. 10. Helmut, V. Groups as Galois groups an introduction, Volume 53, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1996. Received 20 February 2017 Accepted 10 September 2017