Let $(G, \cdot)$ be a group with identity element $e$, let $g \in G$, and let $a,b \in \mathbb{Z}$. For every integer $m \in \mathbb{Z}$, interpret $g^m$ using the standard integer-power convention in a group: positive exponents are repeated products, $g^0=e$, and $g^{-m}=(g^m)^{-1}$ for $m>0$.
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If $\operatorname{ord}(g)=\infty$, meaning that no positive integer $k \in \mathbb{N}$ satisfies $g^k=e$, then $g^a=g^b \iff a=b$.
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If $\operatorname{ord}(g)=n$ for some $n \in \mathbb{N}$, meaning that $n$ is the least positive integer satisfying $g^n=e$, then $g^a=g^b \iff n \mid (a-b)$.