Natural Numbers under Addition do not form Group

Theorem

The algebraic structure $\left({\N, +}\right)$ consisting of the set of natural numbers $\N$ under addition $+$ is not a group.

Proof

From Natural Numbers under Addition form Commutative Monoid, $\left({\N, +}\right)$ has an identity element $0$.

However, for any $x \in \N$ such that $x \ne 0$ there exists no $y \in \N$ such that $x + y = 0$.

Thus the general element of $\left({\N, +}\right)$ has no inverse.

Hence the result by definition of group.

$\blacksquare$