Ballester-Bolinches, A.Beidleman, J. C.Esteban Romero, RamónRagland, M. F.2018-03-262018-03-2620160214-1493https://riunet.upv.es/handle/10251/99724[EN] We call G a Hall_X -group if there exists a normal nilpotent subgroup N of G for which G/N' is an X -group. We call G a T0 -group provided G/\Phi(G) is a T -group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define Hall_X -groups and T_0 -groups where X ∈ {T , PT , PST }; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.Reserva de todos los derechosSubnormal SubgroupT-GroupPST-GroupFinite Solvable Group.MATEMATICA APLICADAArtículo10.5565/PUBLMAT_60116_10Abierto