Products of similar matrices (Q1906773)

Let \(\text{SL}_n(K)\) and \(\text{GL}_n(K)\) denote the special and the general linear group of dimension \(n\) over \(K\), respectively. Assume \(|K|\geq 2n\). The author shows, if \(\Omega\) is a nontrivial similarity class in \(\text{SL}(V)\), then every element in \(\text{SL}(V)\) is a product of \(n + 2\) or fewer elements in \(\Omega\). Further, assume \(\text{char }K\neq 2\) and \(|K|\geq 7\) and \(n\geq 4\). If \(\Phi\) is a nontrivial conjugacy class of \(\text{GL}_n(K)\), then every element in \(\text{SL}_n(K)\) is a product of at most \(n+4\) elements in \(\Phi\Phi^{-1}\).