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16.98 (2006)

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Suppose that $G$ is a solvable finite group and $A$ is a group of automorphisms of $G$ of relatively prime order. Is there a bound for the Fitting height $h(G)$ of $G$ in terms of $A$ and $h(C_G(A))$, or even in terms of the length $l(A)$ of the longest chain of nested subgroups of $A$ and $h(C_G(A))$?

Contributor: A. Turull

Progress

When $A$ is solvable, it is proved in (A. Turull, J. Algebra, 86 (1984), 555–566) that $h(G) \leqslant h(C_G(A)) + 2l(A)$ and this bound is best possible for $h(C_G(A)) > 0$. For $A$ non-solvable some results are in (H. Kurzweil, Manuscripta Math., 41 (1983), 233–305).

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