Rotation group of $20$ degrees

Question

Let $R_{20}$ be a rotation counterclockwise by $20$ degrees in the $xy$ plane. What is this group? Then re-write the group in terms of complex numbers of the form $e^{i\phi}$.

Is their a special name for the group? I know it is just the group generated by $20$ degrees, but is there a special name for this? Also how would I write them in the for of $e^{i\phi}$ What is $\phi$ in this case?

Answer 1

1. Convert 20 degrees to radians.
2. That becomes phi in Euler's identity.

That will give it to you in terms of the complex numbers.

You know 20 degrees is $\frac{\pi}{9}$ radians.

Therefore you will write the complex number as $e^{i\frac{\pi}{9}}$

Now in terms of the group it is $C_{n}$ because it is a counterclockwise rotation.

you know it is 20 degrees and $n$ is calculated by dividing 360 by a number

so $\frac{360}{18}=20$

Therefore this is group $C_{18}$.

Do you need to know all of the groups or just that one in particular?