Generator of Cyclic Group/Examples/Subgroup of Multiplicative Group of Real Numbers Generated by 2
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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:
- $4$ is not a generator of $\gen 2$.
Proof
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 $\gen 2$.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39$. Cyclic Groups