5.30 (1976)
Open(Well-known problem). Suppose that $G$ is a finite soluble group, $A \leqslant \operatorname{Aut} G$, $C_G(A) = 1$, the orders of $G$ and $A$ are coprime, and let $|A|$ be the product of $n$ not necessarily distinct prime numbers. Is the nilpotent length of $G$ bounded above by $n$?
Progress
This is proved for large classes of groups (E. Shult, F. Gross, T. Berger, A. Turull), and there is a bound in terms of $n$ if $A$ is soluble (J. G. Thompson’s $\leqslant 5^n$, H. Kurzweil’s $\leqslant 4n$, A. Turull’s $\leqslant 2n$), but the problem remains open.
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