  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.
