Abstract Algebra

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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:

if the 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.

A

$5$ is a primitive root mod $12$

B

There are only $3$ primitive $12$-th roots of unity

C

$({\mathbb Z}/9{\mathbb Z})^\ast=\langle 2\,{\rm mod}\,9\rangle=\langle 5\,{\rm mod}\,9\rangle$

D

If $h\in\langle g\rangle$ does not generate $\langle g\rangle$ then the order of $h$ does not divide the order of $g$

E

If there is at least one primitive root mod $p$, $p\ge2$ prime, then there are $\varphi(p-1)$ of them.