# Natural Numbers under Addition do not form Group

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## Theorem

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

### Corollary

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

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.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.1$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.3$: Example $3$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): $\S 1$: Some examples of groups

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- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Exercise $1$