Products of strictly hyperbolic conjugacy classes in symplectic groups

TL;DR

The paper proves that, for $2n\ge 4$, the product of two cyclic, strictly hyperbolic conjugacy classes in $Sp(2n,K)$ contains all nonscalar elements, implying that $PSp(2n,K)$ has a conjugacy class of covering number 2 in many cases. The approach hinges on constructing minimal polynomials $q$ with $q$ and its reciprocal $q^*$ coprime, and exploiting principal corners of symplectic conjugacy classes to realize products that generate the whole nonscalar set. A detailed classification of orthogonally indecomposable symplectic transformations (types 1, 2, 3 and subtypes 1e, 1o) underpins the structural analysis, with characteristic-2 and odd-characteristic cases treated separately. This yields explicit polynomial choices for various fields (including finite fields like $\mathrm{GF}(2)$ and $\mathrm{GF}(3)$) and culminates in a verification of Thompson's conjecture in the special case of projective symplectic groups, via the achieved covering-number-2 phenomenon.

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.

Theorems & Definitions (60)

• Theorem 1.1
• Corollary 1.2
• proof
• Proposition 1.3
• Lemma 1.4
• Lemma 1.5
• proof : Proof of \ref{['theorem-1']}
• Remark 2.1
• Definition 2.1
• Lemma 2.2