Finite soluble groups admitting an automorphism of prime power order with few fixed points

Abstract

Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of theCONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.

Finite groups with an almost regular automorphism of order four

“…Clearly, the Sylow 2-subgroup D and the Hall 2 -subgroup E of G/F (G) = D × E may be treated separately. The fact that the index |D : C D (ϕ 2 )| is bounded in terms of m was pointed out (in a more general situation) in [19,Lemma 4·6].…”

“…By Fong's theorem [32], the group G contains a soluble subgroup of m-bounded index; hence we can assume that the group G is soluble. By the Hartley-Turau theorem [19], the index of the second Fitting subgroup |G : F 2 (G)| is bounded in terms of m. Therefore we may suppose that G = F 2 (G).…”

“…A standard argument shows that the order G/O 2 ,2 (G) is bounded in terms of m (see, e.g., [19,Prop. 4·3]): this quotient group acts faithfully on the Frattini quotient group V of O 2 ,2 (G)/O 2 (G), and the rank of V is at most 4m in view of the structure of the Jordan normal form of a linear transformation ϕ (since the number of fixed points of ϕ on V is at most m by (9)).…”

“…Nevertheless, that condition is missing in the general theorem of Dade, which gives bounds for the Fitting height of a finite soluble group with a regular nilpotent automorphism group (see [18]). It is also worth mentioning the Hartley-Turau theorem with a sharp bound for a cyclic group of automorphisms of order p n (see [19]). At the same time the problem posed by Hartley and Belyaev in [1,Problem 13.8a], which in essence calls for bounding the Fitting height of a finite soluble group containing an element with centralizer of given order, is still not resolved.…”

A general Brauer-Fowler theorem and centralizers in locally finite groups

“…Taking into account [12], the following may either be deduced from Theorem A, or deduced from Theorem A! by an argument analogous to Brauer-Fong [4], We leave the details to the reader.…”