Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers

Examples of Order of Group Element

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

The order of $2$ in $\struct {\R_{\ne 0}, \times}$ is infinite.

Proof

From Real Multiplication Identity is One, the identity of $\struct {\R_{\ne 0}, \times}$ is $1$.

There exists no $n \in \Z_{\ge 0}$ such that $2^n = 1$.

Hence the result by definition of infinite order element.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 38$. Period of an element: Illustrations: $\text{(i)}$