[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.

The first and the third authors have been supported by the research project MTM2010-19938-C03-01 from the Ministerio de Ciencia e Innovacion, Spain. The first author has also been supported by the National Natural Science Foundation of China (Grant no. 11271085).