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Abstract
The Hurwitz space is the space of genus covers of the Riemann sphere with branch points and the monodromy group . Let be the symmetric group . In this paper, we enumerate the connected components of . Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of . This work gives us the complete classification of primitive genus zero symmetric group of degree seven.
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References
- Volklein, H. Groups as Galois groups An Introduction to Cambridge Studies in Advanced Mathematics, volume 53, Cambridge University Press, Cambridge, 1996.
- Mohammed Salih, H. Finite Groups of Small Genus. 2014, Ph.D. Thesis, University of Birmingham.
- Michael, D.F. and Volklein, H. The inverse Galois problem and rational points on moduli spaces. Mathematical Annual, 1991, 290(4), 771-800.
- Frohardt D., Guralnick, R. and Magaard K. Genus 0 actions of groups of Lie rank 1. In Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), Proceeding Symposium in Pure Mathematics, 2002, 70, 449-483.
- Frohardt, D. and Magaard, K. Composition factors of monodromy groups. American Mathematical Society, Providence, RI, Annals of Mathematics, 2001, 154(2), 327-345.
- Frohardt, D., Guralnick, R. and Magaard, K. Primitive monodromy groups of genus at most two, Journal of Algebra, 2014, 417, 234-274.
- The GAP Group. GAP-Groups, Algorithms, and Programming, Version 4.6.2, 2013.
- Guralnick, R and John, G. Thompson. Finite groups of genus zero. Journal of Algebra, 1990, 131(1), 303-341.
- Magaard, K., Shpectorov, S. and Wang, G. Generating sets of affine groups of low genus. In Computational algebraic and analytic geometry, Contemporary Mathematics, American Mathematical Society, Providence, RI, 2012, 572, 173-192.
- Mohammed Salih, H. Connected components of H_r^in (G). Sultan Qaboos University Journal for Science, 2017, 22(2), 106-113.
- Neubauer, M. On solvable monodromy groups of fixed genus. 1989. Ph.D. Thesis, University of Southern California.
- Guralnick, R. and John, G.T. Finite groups of genus zero. Journal of Algebra, 1990, 131(1), 303-341.
References
Volklein, H. Groups as Galois groups An Introduction to Cambridge Studies in Advanced Mathematics, volume 53, Cambridge University Press, Cambridge, 1996.
Mohammed Salih, H. Finite Groups of Small Genus. 2014, Ph.D. Thesis, University of Birmingham.
Michael, D.F. and Volklein, H. The inverse Galois problem and rational points on moduli spaces. Mathematical Annual, 1991, 290(4), 771-800.
Frohardt D., Guralnick, R. and Magaard K. Genus 0 actions of groups of Lie rank 1. In Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), Proceeding Symposium in Pure Mathematics, 2002, 70, 449-483.
Frohardt, D. and Magaard, K. Composition factors of monodromy groups. American Mathematical Society, Providence, RI, Annals of Mathematics, 2001, 154(2), 327-345.
Frohardt, D., Guralnick, R. and Magaard, K. Primitive monodromy groups of genus at most two, Journal of Algebra, 2014, 417, 234-274.
The GAP Group. GAP-Groups, Algorithms, and Programming, Version 4.6.2, 2013.
Guralnick, R and John, G. Thompson. Finite groups of genus zero. Journal of Algebra, 1990, 131(1), 303-341.
Magaard, K., Shpectorov, S. and Wang, G. Generating sets of affine groups of low genus. In Computational algebraic and analytic geometry, Contemporary Mathematics, American Mathematical Society, Providence, RI, 2012, 572, 173-192.
Mohammed Salih, H. Connected components of H_r^in (G). Sultan Qaboos University Journal for Science, 2017, 22(2), 106-113.
Neubauer, M. On solvable monodromy groups of fixed genus. 1989. Ph.D. Thesis, University of Southern California.
Guralnick, R. and John, G.T. Finite groups of genus zero. Journal of Algebra, 1990, 131(1), 303-341.