Let $\Bbb Z$ be the additive group of integers and $S = \{-1,1\}$ be a group under multiplication. Is the product $\Bbb Z \times S$ cyclic? Why or why not?
I am really confused on this question and have no idea where to start.
2026-03-26 07:58:18.1774511898
Direct product of groups is cyclic or not?
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To be a cyclic group, there needs to be a generator: That is, there needs to be an element $g$ of the group such that $\langle g \rangle = G$, where $\langle g \rangle$ is the subgroup generated by $G$.
Suppose that we had a generator $g = (z, s)$ with $z \in \mathbb{Z}$ and $s \in S$. Can you deduce that $z$ must be either $-1$ or $1$? If so, can you see the four cases to test as possible generators? Then can you conclude that there are, in fact, no generators?