On the length of finite groups and of fixed points

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. 3781-3790. ISSN 0002-9939

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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.
1

Additional Information:MathSciNet review: 3359570
Keywords:Frobenius group, Automorphism, Nilpotency class, Lie ring, Finite p-group, bmjgoldcheck, NotOAChecked
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:16206
Deposited On:05 Dec 2014 15:08

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