A probabilistic version of a theorem of Kegel. (Q2487029)

\textit{O. H. Kegel} proved [Math. Z. 75, 373-376 (1961; Zbl 0104.24904)] that if a finite group \(G\) admits an automorphism \(\alpha\) of prime order \(p\) such that \(xx^\alpha x^{\alpha^2}\cdots x^{\alpha ^{p-1}}=1\) for all \(x\in G\), then \(G\) is nilpotent. The author proves that it is actually sufficient to require that this operator identity holds with sufficiently high probability depending only on \(p\) (and gives a sharp bound for this probability).