Formulating basic notions of finite group theory via the lifting property

We reformulate several basic notions of notions in finite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions being nilpotent, solvable, perfect, torsion-free; p-groups and prime-to-p-groups; Fitting subgroup, perfect core, p-core, and prime-to-p core. We also reformulate as in similar terms the conjecture that a localisation of a (transfinitely) nilpotent group is (transfinitely) nilpotent.