Proportions of \(r\)-regular elements in finite classical groups. (Q2843988)

For each prime \(r\) let \(p_r(G)\) denote the proportion of elements in a finite group \(G\) which are \(r\)-regular. Bounds on \(p_r(G)\), particularly lower bounds for the simple or almost simple groups, have applications in computational group theory. Some bounds are already known; for example, \textit{L. Babai, P. P. Pálfy} and \textit{J. Saxl} [LMS J. Comput. Math. 12, 82-119 (2009; Zbl 1225.20013)] show that \(p_r(G)\geq\frac{1}{2d}\) for any classical linear group of degree \(d\).NEWLINENEWLINE The main theorems of the present paper generalize and improve the known results. (Theorem 1.1): (a) if \(G=\mathrm{PSL}_d(q)\) or \(\mathrm{PSU}_d(q)\), then \(p_r(G)\geq 1/d\); (b) if \(G=\mathrm{Sp}_{2n}(q)\), \(\mathrm{SO}_{2n}^\pm(q)\), \(\mathrm{SO}_{2n+1}^o(q)\), \(\Omega_{2n}^\pm(q)\) or \(\Omega_{2n+1}^o(q)\) or one of the corresponding projective groups then \(p_r(G)\geq\frac{50}{87}(\pi n)^{-1/2}\) if \(r\) is odd, \(p_2(G)\geq\frac{1}{4}(n+1)^{-3/4}\) if \(G=\mathrm{SO}_{2n}^\pm(q)\) and \(p_2(G)\geq\frac{1}{8}(n+1)^{-3/4}\) if \(G\neq\mathrm{SO}_{2n}^\pm(q)\).NEWLINENEWLINE Theorem 1.2 gives corresponding upper bounds which indicate that in many cases the lower bounds are not far from being asymptotically best possible. Other theorems give further refinements such as, for example, Theorem 1.5 which shows that for each \(\varepsilon>0\) there exists a constant \(C\) such that \(p_2(\mathrm{PSL}_d(q))\leq Cd^{\varepsilon-1}\) for infinitely many prime powers \(q\) and all \(d\geq 2\).