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.
 A: 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:

Pakianathan, Jonathan. Shankar, Krishnan.  Nilpotent Numbers. Amer. Math. Monthly, Vol. 107, No. 7, 631-634 (2000).

References for the cyclic case can be found in this question. See also the answer by Pete L. Clark in here.
