17.119 (2010)
OpenSuppose that a finite soluble group $G$ admits a soluble group of automorphisms $A$ of coprime order such that $C_G(A)$ has rank $r$. Let $|A| $ be the product of $l$ not necessarily distinct primes. Is there a linear function $f$ such that $G/F_{f(l)}(G)$ has $(|A|, r)$-bounded rank, where $F_{f(l)}(G)$ is the $f(l)$-th Fitting subgroup?
Progress
An exponential function $f$ with this property was found in (E. Khukhro, V. Mazurov, Groups St. Andrews 2005, vol. II, Cambridge Univ. Press, 2007, 564–585). It is also known that $|G/F_{2l+1}(G)|$ is bounded in terms of $|C_G(A)|$ and $|A|$ (Hartley–Isaacs).
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