Journal of Algebra, vol.643, pp.1-10, 2024 (SCI-Expanded)
Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ(A), the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2ℓ(A) if |G| is odd. More generally we prove that the nilpotent length of G is at most 2ℓ(A)+c(G;A) when G is of odd order and A normalizes a Sylow system of G where c(G;A) denotes the number of trivial A-modules appearing in an A-composition series of G.