What's so special about finite presentations?
Does it indicate some properties about the group?
A finitely presented group is countable (and I assume finite is countable). In the grand scheme of things being countable is an exceptionally rare event for a group.
On a practical scale, they're about the only things one can compute with, though even then there are famous conjectures about how hard it is to determine something about a group from a presentation.
They're also the groups that keep cropping up 'in nature'. Nature in the case can be taken to mean 'topology'. E.g. homotopy groups will occur naturally as finitely presented groups. Take the torus for example, it is standard to write/define/discover the fundamental group as generated by two loops going in the different directions around a torus, subject to the relation that they commute (this makes the group ZxZ).
Finite is always countable. Isn't that obvious?
Can you list some of the conjectures?
Some people use the word 'countable' to mean what you would call 'countably infinite' -- i.e. to mean bijective with the natural numbers. Matt looks like he was simply explicitly stating how he is using the word.
It's a convention that not all people adopt, so no it isn't at all obvious. Google for
'conjectures finitely presented groups'
to get some ideas. Probably the most famous is 'the word problem': identify when some word (i.e. some string of generators) is equal to the identity (or when two words are equal). Even 'is the group finite' is hard.
The hardness of these things might suggest that looking at groups via their presentations alone might not be a good way to study the groups. In other words the presentation definitions is that useful/good?
OK, so I give you a group G. I tell you nothing about it at all. How are you going to prove anything? A finite presentation actually allows you to make deductions about the group. It's just that there are no algorithms that are particularly fast.
Certainly, one tries to do things other than just play with the generators and relations, such as trying to impose a hyperbolic metric on some associated space, for example. Or tries to find an action on something that tells you more.
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