On a Fitting length conjecture without the coprimeness condition. (Q1759666)

Let \(G\) be a finite solvable group, let \(A\) be a nilpotent group of automorphisms of \(G\), and assume that \(C_G(A)=1\). It is conjectured that then the Fitting height \(h(G)\) of \(G\) is bounded above by the length \(\ell(A)\) of the longest chain of subgroups of \(A\). While this is known to be true in many cases, the conjecture is still open in general. Recent results include [\textit{G. Ercan} and \textit{İ. Ş. Güloğlu}, J. Algebra 320, No. 1, 426-436 (2008; Zbl 1155.20021) and \textit{G. Ercan, İ. Ş. Güloğlu} and \textit{Ö. M. Sağdiçoğlu}, Proc. Edinb. Math. Soc., II. Ser. 54, No. 1, 77-89 (2011; Zbl 1216.20009)]. In the present paper, the authors prove this conjecture when \(A\) is cyclic of order \(p^nq\), \(p\) and \(q\) primes coprime to \(6\), and \(G\) has Abelian Sylow \(2\)-subgroups.