On the genus of the groups PSL(2, q), PSL(3, q), and PSp(4, q) |
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| Affiliation: | 1. School of Electronic and Electrical Engineering, Wuhan Textile University, Wuhan,= 430073, PR China;2. School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China;3. Department of Electrical & Electronic Engineering, South University of Science and Technology of China, Shenzhen, 518055, PR China;4. School of Power and Mechanical Engineering, Wuhan University, Wuhan, Hubei 430074, PR China;1. Departamento de Física, Universidad de Extremadura, Badajoz, Spain;2. Instituto Universitario de Investigación del Agua, Cambio Climático y Sostenibilidad (IACYS), Universidad de Extremadura, Badajoz, Spain;3. Departamento de Física Aplicada, Universidad de Granada, Granada, Spain;4. Instituto Pirenaico de Ecología, Consejo Superior de Investigaciones Científicas (IPE-CSIC), Zaragoza, Spain;5. Departamento de Física, Universidad de Girona, Girona, Spain;6. Andalusian Institute for Earth System Research (IISTA-CEAMA), University of Granada, Autonomous Government of Andalusia, Granada, Spain |
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| Abstract: | SL(n, q) is the group of n×n matrices, over the Galois field GF(q), of determinate 1. PSL(n, q) is SL(n, q) modulo the scalar n×n matrices of determinate 1. PSL(n, q) acts on the Desarguesian projective space PG(n−1, q). Sp(4, q) is the group of 4 × 4 matrices of determinate 1 which preserve the symplectic bilinear form on the 4 × 1 matrices over GF(q). PSp(4, q) is Sp(4, q) modulo Z = {1,−1}. PSp(4, q) acts on the symplectic generalized quadrangle W(3, q), a subspace of the projective space PG(3, q), as a group of automorphisms. In this paper, bounds are given for the genus of these groups. |
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