2015/05/31 by Khukhro, Evgeny, Shumyatsky, Pavel
#20D25 #20D45 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1506.00233
Let g be an element of a finite group G. For a positive integer n, let En(g) be the subgroup generated by all commutators [...[[x,g],g],…,g] over x∈ G, where g is repeated n times. By Baer's theorem, if En(g)=1, then g belongs to the Fitting subgroup F(G). We generalize this theorem in terms of certain length parameters of En(g). For soluble G we prove that if, for some n, the Fitting height of En(g) is equal to k, then g belongs to the (k+1)th Fitting subgroup Fk+1(G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h^*(H) of a finite group H is the least number h such that F^*h(H)=H, where F^*0(H)=1, and F^*i+1(H) is the inverse image of the generalized Fitting subgroup F^*(H/F^*i(H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of En(g) is equal to k, then g belongs to F^*f(k,m)(G), where f(k,m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(En(g))=k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.