On the characterization of the numbers $n$ such that any group of order $n$ has a given property $P$

TL;DR

This paper surveys classical results in group theory concerning the characterization of positive integers n for which all groups of order n possess certain properties like cyclic, abelian, or solvable, providing a unified exposition of known solutions.

Contribution

It synthesizes and presents known results and tools used to determine integers n for which groups of order n have specific properties, serving as a comprehensive reference.

Findings

Compilation of solutions for cyclic, abelian, nilpotent, and supersolvable groups

Presentation of Sylow theorems and solvable groups in this context

Unified account of methods used in classical group theory problems

Abstract

One of the classical problems in group theory is determining the set of positive integers $n$ such that every group of order $n$ has a particular property $P$, such as cyclic or abelian. We first present the Sylow theorems and the idea of solvable groups, both of which will be invaluable in our analysis. We then gather various solutions to this problem for cyclic, abelian, nilpotent, and supersolvable groups, as well as groups with ordered Sylow towers. This work is an exposition of known results, but it is hoped that the reader will find useful the presentation in a single account of the various tools that have been used to solve this general problem. This article claims no originality, but is meant as a synthesis of related knowledge and resources.