Abstract

Abstract. 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.
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