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16.99 (2006)

Open

Suppose that $G$ is a finite solvable group, $A \leqslant \text{Aut}\,G$, $C_G(A) = 1$, the orders of $G$ and $A$ are coprime, and let $l(A)$ be the length of the longest chain of nested subgroups of $A$. Is the Fitting height of $G$ bounded above by $l(A)$?

Contributor: A. Turull

Progress

For $A$ solvable the question coincides with 5.30. It is proved that for any finite group $A$, first, there exist $G$ with $h(G) = l(A)$ and, second, there is a finite set of primes $\pi$ (depending on $A$) such that if $|G|$ is coprime to each prime in $\pi$, then $h(G) \leqslant l(A)$. See (A. Turull, Math. Z., 187 (1984), 491–503).

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