# Finite presentation of groups

What's so special about finite presentations?

Does it indicate some properties about the group?

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matt grime
Homework Helper
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).

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.
Finite is always countable. Isn't that obvious?

Can you list some of the conjectures?

Hurkyl
Staff Emeritus
Gold Member
Finite is always countable. Isn't that obvious?
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.

matt grime
Homework Helper
Finite is always countable. Isn't that obvious?
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.

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?

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.
Right.

matt grime