On the Length of Finite Groups and of Fixed Points

The generalized Fitting height of a finite group G is the least number h = h∗(G) such that F ∗ h (G) = G, where the F ∗ i (G) is the generalized Fitting series: F ∗ 1 (G) = F ∗(G) and F ∗ i+1(G) is the inverse image of F ∗(G/F ∗ i (G)). It is proved that if G admits a soluble group of automorphisms A of coprime order, then h∗(G) is bounded in terms of h∗(CG(A)), where CG(A) is the fixed-point subgroup, and the number of prime factors of |A| counting multiplicities. The result follows from the special case when A = 〈φ〉 is of prime order, where it is proved that F ∗(CG(φ)) F ∗ 9 (G). The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series each of whose factors is either soluble or is a direct product of nonabelian simple groups. It is proved that if A is a group of automorphisms of G of coprime order, then λ(G) is bounded in terms of λ(CG(A)) and the number of prime factors of |A| counting multiplicities.