Products of strictly hyperbolic conjugacy classes in symplectic groups

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 $q(x)q^{\ast }(x)$, where the polynomial $q(x)$ is prime to its reciprocal $q^{\ast }(x):=x^{n}q(x^{-1})$. 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$(2n,K)={\Omega }^2$ for some conjugacy class $\Omega$ of PSp$(2n,K)$. This verifies a conjecture of J. G. Thompson in the special case of a (finite) projective symplectic group.

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