On the Fitting length of \(H_ n(G)\) (Q1313436)

The authors prove the following theorem: Let \(H\) be a finite solvable group, \(\alpha\) an automorphism of \(H\) of prime order acting on \(H\) in such a way that \(p^ 2 \nmid N\) for \(N := \text{lcm}\{o(a);\;a\in H\langle \alpha \rangle \setminus H\}\). If \(4 \nmid N\) then the Fitting length of \(H\) is at most \(m + 4\), \(m\) being the number of prime factors of \(N/p\). Applied to the Hughes subgroups \(H_ n(G) = \langle x \in G; x^ n \neq 1\rangle\) of a finite solvable group \(G\) this result implies that their Fitting length is at most \(m + 3\) if \(H_ n(G) \neq G\) and \(n = p_ 1 \cdot \dots \cdot p_ m\) with pairwise distinct primes \(p_ i\).