Genus zero of projective symplectic groups
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Abstract
A transitive subgroup G ≤ SN is called a genus zero group if there exist non identity elements x1 , . . . , xr∈G satisfying G =<x1, . . . , xr>, x1·...·xr=1 and ind x1+...+ind xr = 2N − 2. The Hurwitz space Hinr(G) is the space of genus zero coverings of the Riemann sphere P1 with r branch points and the monodromy group G.
In this paper, we assume that G is a finite group with PSp(4, q) ≤ G ≤ Aut(PSp(4, q)) and G acts on the projective points of 3-dimensional projective geometry PG(3, q), q is a prime power. We show that G possesses no genus zero group if q > 5. Furthermore, we study the connectedness of the Hurwitz space Hinr(G) for a given group G and q ≤ 5.
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How to Cite
Mohammed Salih, H., & Rezhna M. Hussein, R. M. (2022). Genus zero of projective symplectic groups. Extracta Mathematicae, 37(2), 195-210. https://doi.org/10.17398/2605-5686.37.2.195
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Section
Group Theory
References
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[3] R.M. Guralnick, J. Thompson, Finite groups of genus zero, J. Algebra 131 (1) (1990), 303 – 341.
[4] X. Kong, Genus 0, 1, 2 actions of some almost simple groups of lie rank 2, PhD Thesis, Wayne State University, 2011.
[5] K. Magaard, Monodromy and sporadic groups, Comm. Algebra 21 (12) (1993), 4271 – 4297.
[6] H.M. Mohammed Salih, Hurwitz components of groups with socle PSL (3, q), Extracta Math. 36 (1) (2021), 51 – 62.
[7] L.L. Scott, Matrices and cohomology, Ann. of Math. (2) 105 (3) (1977), 473 – 492.