Difficult# Primitive Roots mod m and Cyclic Subgroups

ABSALG-UTGNAI

Let $G$ be a group. For $g\in G$, let:

$\langle g\rangle=\{g^n\mid n\in{\mathbb Z}\}$

...be the cyclic subgroup of $G$ generated by $g$.

Instead of the word *generator*, we often use *primitive*. For example:

ifthe group $({\mathbb Z}/m{\mathbb Z})^\ast$ of integers mod $m$ coprime to $m$ is cyclic, a generator is called a primitive root mod $m$

...For another example:

a complex number $\zeta$ such that $m$ is the smallest positive integer with $\zeta^m=1$ is called a primitive $m$-th root of unity.

Which of the following are **CORRECT**?

*You do not need a calculator.*

For **Choice 'E'**, if $m$ is a positive integer $\varphi(m)$ denotes the value of Euler's Phi Function at $m$.

Select **ALL** that apply.