Formulating basic notions of finite group theory via the lifting property

TL;DR

This paper redefines fundamental finite group theory concepts using the lifting property, providing a new categorical perspective on properties like nilpotency, solvability, and various subgroup structures.

Contribution

It introduces a novel formulation of key finite group notions through the lifting property, unifying diverse concepts under a categorical framework.

Findings

Reformulation of nilpotent, solvable, and perfect groups via lifting property.

Representation of p-groups and related structures categorically.

Conjecture on localizations preserving transfinite nilpotency.

Abstract

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.