Which of the following groups are cyclic:
- $\langle \mathbb{Z}, +\rangle$
- $\langle \mathbb{Q}, +\rangle$
- $\langle\mathbb{Q^+}, \cdot\rangle$
- $\langle \mathbb{6Z}, +\rangle$
- $\langle \{6^n\mid n \in \mathbb{Z}\}, \cdot\rangle$
- $\langle \{a + b \sqrt{2} \mid a,b \in \mathbb{Z}\}, +\rangle$
I was typing out my answers to these when I realized I had no idea what I was doing, despite prior efforts. So any explanation of why these are cyclic or not cyclic is appreciated.
I know a group is cyclic if a generator can create the entire group, but I have no idea what I'm doing with regards to the actual generators.
Thanks for any help!
Reference: Fraleigh p. 56 Question 5.26 (sorry for typo) in A First Course in Abstract Algebra