# Natural Numbers under Addition do not form Group

## Theorem

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

## Corollary to Natural Numbers under Addition do not form Group

The algebraic structure $\struct {\Z_{\ge 0}, +}$ consisting of the set of non-negative integers $\Z_{\ge 0}$ under addition $+$ does not form a subgroup of the additive group of integers.

## Proof

By Natural Numbers are Non-Negative Integers, $\struct {\Z_{\ge 0}, +}$ and $\struct {\N, +}$ are the same (or if not exactly the same, at least isomorphic).

The result follows from Natural Numbers under Addition do not form Group.

$\blacksquare$

## Proof

From Natural Numbers under Addition form Commutative Monoid, $\struct {\N, +}$ 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 $\struct {\N, +}$ has no inverse.

Hence the result by definition of group.

$\blacksquare$