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One of the classical problems in group theory is determining the set ofpositive integers $n$ such that every group of order $n$ has a particularproperty $P$, such as cyclic or abelian. We first present the Sylow theoremsand the idea of solvable groups, both of which will be invaluable in ouranalysis. We then gather various solutions to this problem for cyclic, abelian,nilpotent, and supersolvable groups, as well as groups with ordered Sylowtowers.  This work is an exposition of known results, but it is hoped that the readerwill find useful the presentation in a single account of the various tools thathave been used to solve this general problem. This article claims nooriginality, but is meant as a synthesis of related knowledge and resources.

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