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

Example of Generators 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:

$\dfrac 1 2$ is also a generator of $\gen 2$

but:

We have that:

$\dfrac 1 2 = 2^{-1}$

and so:

$\dfrac 1 2 \in \gen 2$

Now let $x \in \gen 2$.

Then $x = 2^m$ for some $m \in \Z$.

It follows that:

$x = \paren {\dfrac 1 2}^{-m}$

for $-m \in \Z$.

Similarly, consider $\gen {\dfrac 1 2}$.

Let $y \in \gen {\dfrac 1 2}$.

Then $y = \paren {\dfrac 1 2}^{-r}$ for some $r \in \Z$.

It follows that:

$y = 2^{-r}$

for $-r \in \Z$.

Thus it is seen that:

$\gen 2 = \gen {\dfrac 1 2}$

and so $\dfrac 1 2$ is a generator of $\gen 2$.

From the argument in Element of Cyclic Group is not necessarily Generator it is seen that $4$ is not a generator of $\gen 2$.

$\blacksquare$