Generator of Subgroup/Examples/Positive Odd Numbers

Example of Generator of Subgroup

Let $A$ be the set of positive odd integers.

Let $\struct {\Z, +}$ be the additive group of integers.

The subgroup of $\struct {\Z, +}$ generated by $A$ is $\struct {\Z, +}$ itself.

Proof

From Generator of Subsemigroup: Positive Odd Numbers, the subsemigroup of $\struct {\Z, +}$ generated by $A$ is the semigroup of strictly positive integers under addition.

Then the set of integers $\Z$ is generated by $\struct {\Z, +}$ by various routes.

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings