List all generators of the set $\{5^n\mid n \in \mathbb{Z}\}$ under multiplication.

Yes. More generally: if a subset $S$ of a group $G$ spans a subgroup $H$, then the set $\{g^{-1}\mid g\in S\}$ also spans $H$.


Yes, $5$ and $\frac{1}{5}$ are both generators. You can also prove there are no more generators. If $|n|>1$ then $5^n$ can't generate the group since there is no $m\in\mathbb{Z}$ such that $5=(5^{n})^m$.

Tags:

Abstract Algebra

Group Theory

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