I'm trying to understand how a given presentation of a group is well defined. It says on Wikipedia (https://en.wikipedia.org/wiki/Presentation_of_a_group) that $G \ =\ \langle a\ |\ a^n=1\rangle$ is a presentation for the cyclic group of order $n$. But to me, that presentation isn't well defined. We're not given that the order of $a$ is $n$, just that $a^n=1$. If $n=8$ say then the trivial group, the cyclic group of order $2$ and the cyclic group of order $4$ also satisfy the conditions of the presentation. How is it that $G$ is actually uniquely defined?
Also, I don't know anything about free groups, quotient groups, normal subgroups or any of that stuff that Wikipedia uses to define group presentations. I'm self-studying Dummit and Foote and they introduce group presentations early on without any of those concepts.