21.67 (2026)
OpenSuppose that $\phi$ is an automorphism of a finite soluble group $G$, and let $r$ be the (Prüfer) rank of the fixed-point subgroup $C_G(\phi)$. Is the Fitting height of $G$ bounded terms of $|\phi|$ and $r$?
An affirmative answer to this question would imply an affirmative answer to 13.8(b).
Progress
This is known to be true in the case where $|\phi|$ is a product of at most two prime powers (B. Hartley, Preprint, 1994, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2025/08/hartley94-prepr-mims.pdf).
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