HALL CLASSES OF GROUPS WITH A LOCALLY FINITE OBSTRUCTION

A well-known theorem of Philip Hall states that if a group G has a nilpotent normal subgroup N such that G/N' is nilpotent, then G itself is nilpotent. We say that a group class X is a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that G/N' belongs to X. Hall classes have been considered by several authors, such as Plotkin ['Some properties of automorphisms of nilpotent groups', Soviet Math. Dokl. 2 (1961), 471-474] and Robinson ['A property of the lower central series of a group', Math. Z. 107 (1968), 225-231]. A further detailed study of Hall classes is performed by us in another paper ['Hall classes of groups', to appear] and we also investigate the behaviour of the class of finite-by-Y groups for a given Hall class Y ['Hall classes in linear groups', to appear]. The aim of this paper is to prove that for most natural choices of the Hall class Y, also the classes (LF)Y and BY are Hall classes, where LF is the class of locally finite groups and B is the class of locally finite groups of finite exponent.