Generators of Additive Group of Integers

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): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem
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