Search

Let G be a finite soluble group, and let h ( G ) be the Fitting length of G . If φ is a fixed-point-free automorphism of G , that is C G ( φ ) = { 1 } , we denote by W ( φ ) the composition length of 〈 φ 〉 . A long-standing conjecture is that h ( G ) ≤ W ( φ ) , and it is known that this bound is always true if the order of G is coprime to the order of φ . In this paper we find some bounds to h ( G ) in function of W ( φ ) without assuming that ( | G | , | φ | ) = 1 . In particular we prove the validity of the “universal” bound h ( G ) < 7 W ( φ ) 2 . This improves the exponential bound known earlier from a special case of a theorem of Dade. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a group and a group of automorphisms of G. An -orbit of G is a set of the form , where g is an element of G. The aim of this paper is to prove that if is abelian and G is a union of a finite number of -orbits then G admits a normal abelian subgroup of finite index. This result answers affirmatively a question raised by Neumann and Rowley (1998) in . [Copyright &y& Elsevier]

The article discusses integral theorems on the actions of abelian groups on groups of automorphisms. It illustrates the notation and preliminary results of these theorems with the use of various algebraic formulas to prove that it is possible to omit the hypothesis that abelian groups on groups of automorphisms is fixed-point-free (FPF).