Cyclic Group/Examples/Subgroup of Multiplicative Group of Real Numbers Generated by 2
Jump to navigation
Jump to search
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: $(1)$