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.

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    $\begingroup$ 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. $\endgroup$ – Daniel Schepler May 3 '18 at 23:18
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    $\begingroup$ 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 $1$. $\endgroup$ – Tobias Kildetoft May 4 '18 at 6:20
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    $\begingroup$ For an easy example of the above, consider the group $\langle a,b\mid a^2 = 1, b^3 = 1, ab = 1\rangle$. $\endgroup$ – Tobias Kildetoft May 4 '18 at 6:22

Roughly speaking, the main idea is that the group defined by a presentation is the "largest" or "freest" group that satisfies the presentation.

In particular, a group satisfying the presentation does not mean that it is the presented group. (It only means that it is a quotient of the presented group.)

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 $n$.

To make this more formal (including the fact that this really defines a unique group), you need a to learn a bit about free groups and so on.

See also: Is a group defined by its generator set and relations?

(In fact, one could argue this is a duplicate.)

  • $\begingroup$ Thanks, that really clears it up. $\endgroup$ – Evan Naugler May 4 '18 at 0:07

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