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So as to get rid of a grey unattached link, I created a stub for finitely presented group.
It’s certainly the definition that one will find in the literature, but I wonder if I’m alone in finding the grammar a little odd here. According to this definition, there is no difference in meaning between a “finitely presented group” and a “finitely presentable group”. But if it were up to me, I would refer to a finitely presented group only if I had a specific finite presentation in mind (a particular diagram exhibiting $G$ as a coequalizer of a pair of morphisms between finitely generated free groups).
I agree with Todd. On the other hand, that’s what we get if we interpret “there is” in the propositions-as-types manner as a $\Sigma$… (-:
In fact you both mirror the thought that came to me when I typed it out! I would also like to ask whether you feel that the isomorphism from the quotient group to $G$ should be part of the definition of a presentation. My preference is that it should be but that then it can be set aside.
I have made a change to the entry. Try the new one for size!
Re #4, of course it should be! I would tend to phrase it as “a presentation of $G$ is a coequalizer diagram $F R \rightrightarrows F X \twoheadrightarrow G$”.
That sounds about right. As Ronnie and Johannes Huebschmann found in their paper on identities among relations, you need that to make sense of the identities.
I have edited the entry accordingly.
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