Year: 2025
Type: preprint
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 $q^*(x) := x^n q(x^{-1})$. 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$(2n,K) = Ω^2$ 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.
Source: arXiv (Cornell University)
Author: Nielsen, Klaus
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7084141328
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