What are the situations, in which any group of order n is abelian

What are the situations in which any, or particular type group of order n, is abelian ?

For example:

Group of order $p^2$ is abelian. where p is prime.

Solution 1:

Every group of order $n$ is abelian if and only if $n$ has prime factorization of the form $$n = p_1^{a_1} p_2^{a_2} \ldots p_t^{a_t}$$

where $a_i \leq 2$ for all $i$ and $\gcd(p_i^{a_i} - 1, p_j) = 1$ for all $i$ and $j$.

This an old result (1905) and due to L. E. Dickson. Reference:

Dickson, L. E. Definitions of a group and a field by independent postulates, Trans. Amer. Math. Soc. Vol. 6, No. 2, 198-204 (1905).

You might be interested in a similar problem for other group properties besides being abelian. Similar characterizations are known for those $n$ for which every group of order $n$ is

cyclic
abelian
nilpotent
nilpotent of class at most $c \in \mathbb{Z}_+ \cup \{\infty\}$
solvable
supersolvable
metabelian
metacyclic
etc..

Worth mentioning is the case of cyclic groups, which is a cute result: every group of order $n$ is cyclic if and only if $\gcd(n, \varphi(n)) = 1$, where $\varphi$ is the Euler totient function. Also, some of these are far from trivial: characterizing those $n$ for which every group is solvable requires Thompson's classification of minimal simple groups. The result is also needed in the case of metabelian groups.

A good starting point for studying the easier results (cyclic, abelian, nilpotent) is the following paper by Pakianathan and Shankar: