Generation of finite almost simple groups by conjugates. (Q1415350)

Let \(G\) be a finite almost simple group and \(L=F^*(G)\). For \(x\in G\), let \(\alpha(x)\) be the minimal number of \(L\)-conjugates of \(x\) which generate the group \(\langle L,x\rangle\). The authors obtain some upper bounds on \(\alpha(x)\). For example, if \(L\) is a simple classical group of dimension at least 5 and \(x\in\Aut(L)\) then \(\alpha(x)\leq n\), unless \(L=\text{Sp}_n(q)\) with \(q\) even, \(x\) is a transvection and \(\alpha(x)=n+1\) (Theorem 4.2).