The Fitting length of finite soluble groups. II: Fixed-point-free automorphisms
The aim of this paper is the obtain restrictions on the Fitting height of a finite soluble group \(G\) with a fixed-point-free automorphism \(\varphi\), that is, \(\text{C}_G(\varphi)=\{1\}\). We denote by \(h(G)\) the Fitting height of \(G\), by \(\pi(G)\) the set of primes dividing \(|G|\), by \(w(\varphi)\) the cardinality of \(\pi(\langle \varphi\rangle)\) and by \(W(\varphi)\) the composition length of \(\langle \varphi\rangle\) (that is, the number of prime divisors of \(\langle \varphi\rangle\) with multiplicities taken into account). It was conjectured that \(h(G)\leq W(\varphi)\), and it is known that if the order of \(\varphi\) is coprime to \(|G|\), then this conjecture holds. In this paper, the author considers the non-coprime case, in which the hypothesis that the order of \(G\) and \(\varphi\) are coprime is removed. The main result of this paper is Theorem~1.3: If \(w(\varphi)\geq 2\), then \(h(G)<(7w(\varphi)-9)W(\varphi)\leq 7W(\varphi)^2\). Some improvements of this result in some particular cases are obtained. Another result of this paper is Theorem~1.6: If \(G\) is a group and \(\varphi\) is an automorphism of \(G\), \(\gcd(|\text{C}_G(\varphi)|,|\langle\varphi\rangle|)=1\), \(w(\varphi)\geq 2\) and \(h(\text{C}_G(\varphi))=h_0\), then \(h(G)<(8w(\varphi)-10)W(\varphi)+\frac{3}{2}(w(\varphi)-1)w(\varphi)h_0\). For Part I see [\textit{G. Busetto} and the author, Arch. Math. 106, No. 5, 409--416 (2016; Zbl 1350.20015)].
scientific article; zbMATH DE number 6754375
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