# Natural Numbers under Addition do not form Group

Jump to navigation
Jump to search

## 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.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 7$: Example $7.1$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.3$: Example $3$

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