We show that the symplectic groups PSp6(q) are Hurwitz for all q = p^m ≥ 5, with p an odd prime. The result cannot be improved since, for q even and q = 3, it is known that PSp6(q) is not Hurwitz. In particular, n = 6 turns out to be the smallest degree for which a family of classical simple groups of degree n, over F_{p^m}, contains Hurwitz groups for infinitely many values of m. This fact, for a given (possibly large) p, also follows from [9] and [10].
Tamburini Bellani, M. C., Vsemirnov, M., Hurwitz Generation of PSp6(q), <<COMMUNICATIONS IN ALGEBRA>>, 2015; 43 (10): 4159-4169. [doi:10.1080/00927872.2014.939756] [http://hdl.handle.net/10807/68028]
Hurwitz Generation of PSp6(q)
Tamburini Bellani, Maria Clara;
2015
Abstract
We show that the symplectic groups PSp6(q) are Hurwitz for all q = p^m ≥ 5, with p an odd prime. The result cannot be improved since, for q even and q = 3, it is known that PSp6(q) is not Hurwitz. In particular, n = 6 turns out to be the smallest degree for which a family of classical simple groups of degree n, over F_{p^m}, contains Hurwitz groups for infinitely many values of m. This fact, for a given (possibly large) p, also follows from [9] and [10].
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