Can anybody please give me an example of a binary operation under which N forms a group? More generally, how to find some operations to make possibly any set a group?



Making N (set of all positive integers) a group
As explained by Arturo, a simple example of a group structure on **N** is the operation a ⊕ b = f-1(f(a) + f(b)) where f:**N**→**Z** is defined by f(2n) = n and f(2n+1) = -n.


The statement that any nonempty set admits a group structure is equivalent to the Axiom of Choice! This is explained in [this answer](https://mathoverflow.net/questions/12973/does-every-non-empty-set-admit-a-group-structure-in-zf/12988#12988).


