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Abstract Algebra

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Generators of Cyclic Groups

ABSALG-YAE8C4

Let $G$ be a group. If we fix a $g\in G$, the cyclic subgroup $\langle g\rangle=\{g^n\mid n\in{\mathbb Z}\}$ of $G$ is, by definition, the subgroup of elements of the form $g^n$, as $n$ runs over all integers, and where $g\in G$ is fixed. We call $\langle g\rangle$ the cyclic subgroup of $G$ generated by $g$.

By definition, the group $G$ is cyclic if $G=\langle g\rangle$ for some $g\in G$. A cyclic group $C$ is called finite cyclic when its order $|C|$ is finite and infinite cyclic otherwise.

Which of the following are TRUE?

Here, $i=\sqrt{-1}$.

Select ALL that apply.

A

The group $({\mathbb Z}/7{\mathbb Z})^\ast=\{a\,{\rm mod}\,7\mid (a,7)=1\}=\langle 2\,{\rm mod}\,7\rangle$ under multiplication mod $7$.

B

The group $({\mathbb Z}/7{\mathbb Z})^\ast=\{a\,{\rm mod}\,7\mid (a,7)=1\}=\langle 3\,{\rm mod}\,7\rangle$ under multiplication mod $7$.

C

The group $S^1=\{t\in {\mathbb C}\mid |t|=1\}$ under multiplication equals $\langle e^{2\sqrt{2}\pi i}\rangle$.

D

The group $\langle e^{2\sqrt{2}\pi i}\rangle$ under multiplication is infinite cyclic.

E

The group $\langle i\rangle$ under multiplication is finite cyclic of order $3$.