# Uniqueness of group given only group presentation

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.

• Given that free groups in particular are an important special case of a group with a given presentation (just the case where the number of relations is 0), I'm personally not sure what answer we could give before you at least understand free groups. May 3 '18 at 23:18
• It is worth noting that while (as the answer states), this really does define a group uniquely, you are right to be worried about whether the order of $x$ really will be $8$, rather than some divisor of $8$. The reason it ends up being the case here is that there exists a group generated by an element of order $8$, but it is certainly possible to write up a presentation akin to this with relations of the form $a^n = 1$ and $b^m = 1$ (plus some others), and where it turns out that the presented group is trivial, even though neither $n$ nor $m$ was $1$. May 4 '18 at 6:20
• For an easy example of the above, consider the group $\langle a,b\mid a^2 = 1, b^3 = 1, ab = 1\rangle$. May 4 '18 at 6:22

For example, in your case, $G$ should be the "largest" group on a generator called $a$ such that $a^n=1$, not just any group where $a^n=1$ is satisfied, and it turns out that $G$ is the cyclic group of order $n$.