LOCALLY (SOLUBLE-BY-FINITE) GROUPS WITH ALL PROPER INSOLUBLE SUBGROUPS OF FINITE RANK

A group G has finite rank r if every finitely generated subgroup of G is at most r-generator. If F is a class of groups then we let G denote the class of groups G in which every proper subgroup of G is either o f finite rank or in G. We let G denote the class of soluble groups and G(d) the class of soluble groups of derived length at most d, where d is a positive integer. We let Lambda denote the set of closure operat ions {L,R,P,P} and let F denote the Lambda-closure of the class of per iodic locally graded groups. Amongst other results we prove that a sol uble G(d)-group is either of finite rank or of derived length at most d and also that a group in the class F boolean AND (LG) is either lo cally soluble, or has finite rank, or is isomorphic to one of SL(2, F) , PSL (2, F) or Sz(F) for suitable locally finite fields F.