School of Mathematics & Physics, University of Lincoln
The paper by Evgeny Khukhro and Pavel Shumyatsky Engel-type subgroups and length parameters of finite groups has been accepted for publication in Israel Journal of Mathematics. The results of the paper have been obtained in collaboration between Evgeny Khukhro of University of Lincoln and Pavel Shumyatsky of University of Brasilia, with Evgeny’s visits to Brasilia supported by CNPq-Brazil grant within the Brazilian Scientific Mobility Program “Ciências sem Fronteiras”.
Abstract: Let be an element of a finite group . For a positive integer , let be the subgroup generated by all commutators over , where is repeated times. By Baer’s theorem, if , then belongs to the Fitting subgroup . We generalize this theorem in terms of certain length parameters of . For soluble we prove that if, for some , the Fitting height of is equal to , then belongs to the th Fitting subgroup . For nonsoluble the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height of a finite group is the least number such that , where , and is the inverse image of the generalized Fitting subgroup . Let be the number of prime factors of counting multiplicities. It is proved that if, for some , the generalized Fitting height of is equal to , then belongs to , where depends only on and . The nonsoluble length of a finite group 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 , then belongs to a normal subgroup whose nonsoluble length is bounded in terms of and . We also state conjectures of stronger results independent of and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.
Full text: https://arxiv.org/abs/1506.00233