2015/01/14 by Detomi, Eloisa, Shumyatsky, Pavel
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1501.03339
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,xg⟩)≤ k for all x,g∈ G such that ⟨ x,xg⟩ is soluble, then h^*(G) is k-bounded.