Some local properties defining \(\mathcal T_0\)-groups and related classes of groups. (Q5963195)

A group \(G\) is called a Hall\(_{\mathcal X}\)-group if there exists a normal nilpotent subgroup \(N\) of \(G\) for which \(G/N'\) is an \(\mathcal X\)-group (so, according to the famous theorem of P. Hall, Hall\(_{\mathcal N}\)-groups are just the nilpotent groups if \(\mathcal N\) denotes the class of nilpotent groups). The authors consider new classes of groups appearing as Hall\(_{\mathcal X}\)-groups where the classes \(\mathcal X\) are generalizations of \(\mathcal T\)-groups and \(\mathcal{PST}\)-groups.