О производной π-длине π-разрешимой группы

Abstract

A new notion of the derived π-length of a π-soluble group is proposed. The dependence of the derived π-length of a π-soluble group on the structure of Hall π-subgroups are found. In particular, it is proved that the derived π-length of a π-soluble group with abelian Sylow p-subgroup for any p ∈ π coincides with the derived length of a Hall π-subgroup. Also it is established that if 2∉ π then the derived π-length of a π-soluble group with a metabelian Hall π-subgroup is at most 3.