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On the length of a finite group and of its 2-generator subgroups

On the length of a finite group and of its 2-generator subgroups The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* h (G) = G, where F* 1 (G) = F* (G) is the generalized Fitting subgroup, and F* i+1(G) is the inverse image of F* (G/F*i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (⟨x, x g ⟩) ≤ k for allx, g ∈ G such that ⟨x, x g ⟩ is soluble, then h* (G) is k-bounded. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

On the length of a finite group and of its 2-generator subgroups

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References (14)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-016-0191-5
Publisher site
See Article on Publisher Site

Abstract

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* h (G) = G, where F* 1 (G) = F* (G) is the generalized Fitting subgroup, and F* i+1(G) is the inverse image of F* (G/F*i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (⟨x, x g ⟩) ≤ k for allx, g ∈ G such that ⟨x, x g ⟩ is soluble, then h* (G) is k-bounded.

Journal

Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals

Published: Feb 17, 2016

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