Khukhro, E. I. and Shumyatsky, P.
(2015)
On the length of finite groups and of fixed points.
Proceedings of the American Mathematical Society, 143
(9).
pp. 37813790.
ISSN 00029939
16206 On the length of finite groups and of fixed points.pdf   [Download] 

Preview 

PDF
16206 On the length of finite groups and of fixed points.pdf
 Whole Document
185kB 
Item Type:  Article 

Item Status:  Live Archive 

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 fixedpoint 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.
1
Repository Staff Only: item control page