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Which of the following groups are cyclic:

  1. $\langle \mathbb{Z}, +\rangle$
  2. $\langle \mathbb{Q}, +\rangle$
  3. $\langle\mathbb{Q^+}, \cdot\rangle$
  4. $\langle \mathbb{6Z}, +\rangle$
  5. $\langle \{6^n\mid n \in \mathbb{Z}\}, \cdot\rangle$
  6. $\langle \{a + b \sqrt{2} \mid a,b \in \mathbb{Z}\}, +\rangle$

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

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    Please use `\langle` and `\rangle` instead of $<$ and $>$2012-05-19

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