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.
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The research was partially supported by GNSAGA.
The research was supported by CAPES and CNPq.
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Detomi, E., Shumyatsky, P. On the length of a finite group and of its 2-generator subgroups. Bull Braz Math Soc, New Series 47, 845–852 (2016). https://doi.org/10.1007/s00574-016-0191-5
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DOI: https://doi.org/10.1007/s00574-016-0191-5