Which of the following groups are cyclic? For each cyclic group, list all the generators of the group. $G_1 = \langle \mathbb{Z},+\rangle\;\;G_2 = \langle\mathbb{Q}, +\rangle\;\;G_3=\langle\mathbb{Q}^+, \cdot\rangle\;\;G_4 = \langle 6\mathbb{Z}, +\rangle$ $G_5 = \{6^n \mid n\in\mathbb{Z}\} \text{ under multiplication}$ $G_6 =\{a + b \sqrt{2}\mid a, b\in \mathbb{Z}\}\;\;\text{under addition} $
My book says that $G_2$ and $G_3$ aren't cyclic, but it doesn't explain how they arrive to that conclusion. How exactly do you show that the groups aren't cyclic? In other words, how do I show that the group has no generator?