On the Fitting length conjecture (Q1108373)

The longstanding Fitting length conjecture asserts the following: Let A be a solvable group acting on the solvable group G. Assume that \((| A|,| G|)=1\) and \(C_ G(A)=1\). Then the Fitting length of G is at most the composition length of A. The bound is best possible. The conjecture was only known for some supersolvable groups A [see \textit{A. Turull}, J. Reine Angew. Math. 371, 67-91 (1986; Zbl 0587.20017)]. Here it is proved for groups of odd order A satisfying the following: (i) If \(p,q\in \pi (A)\), then \(p\nmid q-1\). (ii) If \(| G|\) is even and \(p\in \pi (A)\), then \(p| 2^ k+1\) for some \(k>1\). Observe that (ii) is slightly stronger than assuming that p is not a Mersenne prime.