Abstract:

As defined by Guralnick and Saxl, given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number $\alpha^{\phantom{S}}_{S}(x)$ is the minimum number of conjugates of $x$ in $\langle x,S\rangle$ that generate a subgroup containing $S$. In this paper, for every sporadic group $S$ other than the Monster and an automorphism $x$ of $S$ of prime order,   we complete the determination of the precise value of $\alpha^{\phantom{S}}_{S}(x)$.