15.65 (2002)

A square matrix is said to be separable if its minimal polynomial has no repeated roots, and cyclic if its minimal and characteristic polynomials are equal. For a matrix group $G$ over a finite field define $s(G)$ and $c(G)$ to be the proportion of separable and of cyclic elements respectively in $G$. For a classical group $X(d, q)$ of dimension $d$ defined over the field with $q$ elements let $S(X; q) := \lim_{d \to \infty} s(X(d, q))$ and $C(X; q) := \lim_{d \to \infty} c(X(d, q))$. Independently G. E. Wall (Bull. Austral. Math. Soc., 60, no. 2 (1999), 253–284) and J. Fulman (J. Group Theory, 2, no. 3 (1999), 251–289) have evaluated $S(\text{GL}; q)$ and $C(\text{GL}; q)$, and have found them to be rational functions of $q$. Are $S(X; q)$ and $C(X; q)$ rational functions of $q$ also for the unitary, symplectic, and orthogonal groups?