TITLE: Why is the set of natural numbers not considered a non-Abelian group?
QUESTION [5 upvotes]: I don't understand why the set of natural numbers constitutes a commutative monoid with addition, but is not considered an Abelian group.

REPLY [2 votes]: Assuming the axiom of choice one can actually endow any set with a group structure. 
To get a group structure for $\mathbb N$ define for example $f: \mathbb Z \to \mathbb N$, $k \mapsto 2|k|$ if $k \leq 0$ and $k \mapsto 2k + 1$ otherwise. Now define addition on $\mathbb N$ as follows: let $n + m := f^{-1}(n) + f^{-1}(m)$. 
Verify that $f$ is a (group) isomorphism, that $0$ is the additive identity and that the additive inverse of $n$ is $f(-f^{-1}(n))$.