# Generators of Additive Group of Integers

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## Theorem

The only generators of the additive group of integers $\struct {\Z, +}$ are $1$ and $-1$.

## Proof

From Integers under Addition form Infinite Cyclic Group, $\struct {\Z, +}$ is an infinite cyclic group generated by $1$.

From Generators of Infinite Cyclic Group, there is only one other generator of such a group, and that is the inverse of that generator.

The result follows.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $5$: Subgroups: Exercise $14$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 25$: Theorem $25.2$