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21.65 (2026)

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Suppose that $\phi$ is an automorphism of a finite soluble group $G$. Must $G$ contain a subgroup of index bounded in terms of $|\phi|$ and $|C_G(\phi)|$ whose Fitting height is bounded
$\qquad$ a) in terms of $|\phi|$?
$\qquad$ b) or even in terms of the composition length of $\langle \phi \rangle$?

Contributor: E. I. Khukhro

Progress

An affirmative answer to part (b) is known when $|\phi|$ is a prime power (B. Hartley–V. Turau), or when $(|G|, |\phi|) = 1$ (A. Turull, B. Hartley–I. M. Isaacs). Also cf. 19.43.

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