Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7

The Hurwitz space Hr (G) is the space of genus g = 0 covers of the Riemann sphere P with r branch points and the monodromy group G. Let G be the symmetric group S7. In this paper, we enumerate the connected components of Hr (S7). Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of Hr (S7). This work gives us the complete classification of primitive genus zero symmetric group of degree seven.

Let be a non-trivial conjugacy class of . Then the set = { 1 , … , } in is called the ramification type of the cover . Note that the trivial conjugacy class contains only the identity element.
In this paper, we classify primitive genus 0 systems for 7 . It is clear that there are seven primitive groups of degree 7. In [1], we classified all those groups except 7 . Now we are going to classify the group 7 by using the computer algebra system GAP. All together give the complete classification of primitive genus 0 groups of degree 7.

L
Braid orbits can be interpreted as saying interesting things about components of the moduli space of curves ℳ [3] and equivalence classes of branched covers of the Riemann sphere ℙ 1 .

Computing Indexes and Labeling Conjugacy Classes
In this paper, we discuss two methods for computing index as follows:

Method one (Via Fixed Points)
Let G be a group acting on a finite set Ω of size . If ∈ , define the index of x by where x has order d.We discussed this method in detail in [2].

Method two (Via Cycle Types)
As we know that is the minimal number of 2-cycles needed to express as a product. We will label the fourteen nontrivial conjugacy classes of S 7 as ATLAS notation by: Table 1. Non trivial conjugacy classes of .

Algorithm
To achieve connected components of ℋ ( ), we need to perform the following steps: Step 1: Select the primitive group 7 by using the GAP code [7] : Primitive Group ( 7, 7 ).
Step 2: Find all ramification types that satisfy equation (3) for given 7 , degree 7 and genus 0.
Step 3: Remove those types which have zero structure constant from the character table of 7 via the following equation.
Step 4: For the remaining types, that pass equation (1) which are called generating types.
Step 5: For the generating types, compute braid orbits by using MAPCLASS package. Now we perform the above steps by using the program described in [12], but with a few modifications to it. That is, we remove the condition of affine type in that program. In this paper we only consider primitive groups.

Conclusion
Here, we compute braid orbits on Nielsen class with the aid of the computer algebra system GAP and MAPCLASS package. A result of the algorithm is that it gives the complete classification of the symmetric group 7 up to braid actions and diagonal conjugations. The computation shows that there are exactly 754 braid orbits of S 7 . As a consequence of Lemma 1.3, we find the connected components ℋ ( 7 ) of 7 -curves , such that = 0. So we have 754 connected components ℋ ( 7 ) of the symmetric group of degree seven.