Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers
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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: $\text{(i)}$