Cyclic Group/Examples/Subgroup of Multiplicative Group of Real Numbers Generated by 2

Example of Cyclic Group

Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

Consider the subgroup $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ generated by $2$.

Then $\gen 2$ is an infinite cyclic group.

Proof

We have that $\gen 2$ is subgroup generated by a single element of $\struct {\R_{\ne 0}, \times}$

By definition, $\gen 2$ is a cyclic group.

By Example: Order of Element of Multiplicative Group of Real Numbers, $2$ is of infinite order.

The result follows by definition of infinite cyclic group.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39$. Cyclic groups: $(1)$