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21.66 (2026)

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Suppose that $A$ is a nilpotent group of automorphisms of a finite soluble group $G$. Is the Fitting height of $G$ bounded in terms of $|A|$ and $|C_G(A)|$?

Contributor: E. I. Khukhro

Progress

An affirmative answer is known when $(|G|, |A|) = 1$ (J. G. Thompson, even for soluble $A$, with improved bounds in subsequent papers of H. Kurzweil, A. Turull, B. Hartley–I. M. Isaacs), when $A$ is cyclic (see 19.43), or when $C_G(A) = 1$ (E. C. Dade). Note that for any non-nilpotent finite group $A$ there are finite soluble groups $G$ of unbounded Fitting height with $C_G(A) = 1$ (S. D. Bell–B. Hartley).

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