Hw5 Problem 5

### Problem 5

Find the order of the cyclic subgroup of the given group listed below.
(a) The subgroup of $\mathbb{Z}_{9}$ generated by $6$.
(b) The subgroup of $V$ generated by $c$.
(c) The subgroup of $U_5$ generated by $\cos \frac{4 \pi }{5} + i \sin \frac {4 \pi }{5}$.

Solution

(a)
Observe that $6+_9 6=3$
Also, $3+_9 6=0$
Thus, $\langle 6 \rangle = \{0, 3, 6\}$
So the order of the subgroup generated by $6$ is $3$.

(b)
Observe that $c*c = e$.
Additionally, $e*c = c$ and $c*e = c$.
Thus, $\langle c \rangle = \{e, c\}$.
So the order of the subgroup generated by $c$ is $2$.

(c)
Observe that $\cos \frac{4 \pi }{5} + i \sin \frac {4 \pi }{5} = e^{4i \pi /5 }$ by Euler's formula.
Repetitively multiplying $e^{4i \pi /5 }$ by itself yields $\langle e^{4i \pi /5 } \rangle = \{ e^{4i \pi /5 }, e^{8i \pi /5 }, e^{2i \pi /5 }, e^{6i \pi /5 }, 1 \}$.
Thus, the order of subgroup of $U_5$ generated by $\cos \frac{4 \pi }{5} + i \sin \frac {4 \pi }{5}$ is $5$.