Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finite presentation of groups

  1. Jul 1, 2008 #1

    tgt

    User Avatar

    What's so special about finite presentations?

    Does it indicate some properties about the group?
     
  2. jcsd
  3. Jul 1, 2008 #2

    matt grime

    User Avatar
    Science Advisor
    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).
     
  4. Jul 4, 2008 #3

    tgt

    User Avatar

    Finite is always countable. Isn't that obvious?

    Can you list some of the conjectures?
     
  5. Jul 4, 2008 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  6. Jul 5, 2008 #5

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  7. Jul 5, 2008 #6

    tgt

    User Avatar

    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?
     
  8. Jul 5, 2008 #7

    tgt

    User Avatar

    Right.
     
  9. Jul 5, 2008 #8

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Finite presentation of groups
  1. Finite Group (Replies: 2)

  2. Finite group (Replies: 3)

  3. Finite groups (Replies: 2)

Loading...