Products of strictly hyperbolic conjugacy classes in symplectic groups

2025/09/30 by Nielsen, Klaus
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2509.26237

Abstract

We call a conjugacy class of the symplectic group Sp(2n, K) over a field K strictly hyperbolic if its minimal polynomial is of the form q(x) q^*(x), where the polynomial q(x) is prime to its reciprocal q^*(x) := xⁿ q(x^-1). It is shown that the product of 2 cyclic, strictly hyperbolic conjugacy classes of Sp(2n, K) contains all nonscalar elements of Sp(2n, K). It follows that the projective symplectic group has a conjugacy class of covering number 2, i.e. PSp(2n,K) = Ω² for some conjugacy class Ω of PSp(2n,K). This verifies a conjecture of J. G. Thompson in the special case of a (finite) projective symplectic group.

Citations

• Bireflectionality in special orthogonal groups
• Character estimates for finite classical groups and the asymptotic Thompson Conjecture

Related

• The semisimple conjugacy classes in the symplectic groups
• Products of involutions in symplectic groups I: bireflections
• On the (2,3)-generation of the finite symplectic groups
• Maximal Unramified Tori in Symplectic Groups
• Products of involutions in symplectic groups over general fields (II)