Hurwitz Generation of PS_p6(q)

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].

Key Words:

• Finite simple groups
• Hurwitz generation

2010 Mathematics Subject Classification:

• Primary: 20G40
• Secondary: 20D05

Notes

Communicated by P. Tiep.