TITLE: Does there exist an $n$ such that all groups of order $n$ are Abelian?
QUESTION [12 upvotes]: I know that all groups of order $\leq$ 5 are Abelian and all groups of prime order are Abelian. Are there any other examples? If so is there something special about the orders of these groups?

REPLY [3 votes]: The terminology you're looking for is:

Definition. A number $n \in \mathbb{N}$ is said to be

cyclic iff all groups of order $n$ are cyclic.
abelian iff all groups of order $n$ are abelian.
nilpotent iff all groups of order $n$ are nilpotent.


Using basic group theory, we see that these are related as follows:

Proposition. For natural numbers, we have: $$\mbox{ prime } \rightarrow \mbox{ cyclic } \rightarrow \mbox{ abelian } \rightarrow \mbox{ nilpotent }$$

A Peter explains, your question is essentially: does there exist an abelian number strictly greater than $5$ that isn't prime? Yes, $15$ does the trick. In fact, $15$ has the stronger property of being a cyclic number strictly greater than $5$ that isn't prime.
You can have a look at this mathoverflow post for more information.