Hw5 Problem 4

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### Problem 4

Which of the following groups are cyclic? For each cyclic group, list all the generators of the group.

(a) $G_1=\langle\mathbb{Z},+\rangle$ (b) $G_2=\{a+b\sqrt{5}|a,b\in\mathbb{Z}\}$ under addition (c) $G_3=\langle\mathbb{Q^+},\cdot\rangle$ (d) $G_4=\langle\mathbb{Q},+\rangle$

(e) $G_5=\langle\mathbb{7Z},+\rangle$ (f) $G_6=\{7^n|n\in\mathbb{Z}\}$ under multiplication

**Solution**

(a), (e), (f) are the cyclic groups.

(a):The generators of $G_1$ are $\langle1\rangle$, $\langle-1\rangle$

(e):The generators of $G_5$ are $\langle7\rangle$, $\langle-7\rangle$

(f):The generators of $G_6$ are $\langle7\rangle$, $\langle7^{-1}\rangle$