2014/05/03 by Khukhro, E. I., Shumyatsky, P.
#20D45 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1405.1946
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))\leqslant 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 either is 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.