Results 1 to 3 of 3

Math Help - Don't know what the group O^+ (1,2) is!

  1. #1
    Newbie
    Joined
    Feb 2008
    Posts
    20

    Don't know what the group O^+ (1,2) is!

    Hi,
    I'm told that symmetries in hyperbolic space is o^+ (1,2) - I'm not sure what that is!
    Thanks,
    Sooz
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Jul 2009
    Posts
    2
    I'm assuming o is a flip and r is a rotation. Meaning:

    r( 123 ) → 231
    o( 123) → 321

    The great thing about it is that this is the set of symmetries of a triangle. r represents a rotation, o represents a flip over an altitude of the triangle.

    See: http://en.wikipedia.org/wiki/Dihedral_gr

    o and r are the generators of the group:
    http://en.wikipedia.org/wiki/Generating_

    So to find these inverses, you basically just do what you did, backwards, in the opposite order.

    I'm using ' as the inverse symbol.
    Start with the following:
    o = e
    r = e
    o' = o
    r' = r
    r ' = r
    e is the identity

    This is because o has order 2 and r has order 3. See:
    http://en.wikipedia.org/wiki/Order_%28gr

    Using those rules, we get:

    (or)' = r'o' = r'o = ro

    o'r' = or' = or

    r'o' = ro

    What's nice about these is that if you rotate, then flip, you could have flipped, then rotated in the opposite direction. Does that make sense? So:

    (or) = oror = oor'r = oe = o = e

    You can test that out if you don't believe it:

    123 → 321 → 213 → 312 → 123

    And finally:

    or = er = r
    herpes testing
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Sooz View Post
    Hi,
    I'm told that symmetries in hyperbolic space is o^+ (1,2) - I'm not sure what that is!
    I think it should probably be a capital O (for orthogonal). Then the group of symmetries of hyperbolic space, O^+ (1,2), ought to mean the group of 33 matrices with positive determinant (that's what the + is for) that preserve the quadratic form x_1^2-x_2^2-x_3^2 (which has 1 positive and 2 negative terms).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Order of Group. Direct Product of Cyclic Group
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 19th 2011, 01:06 PM
  2. Obtaining a larger group from a given group
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 14th 2011, 11:32 AM
  3. Replies: 1
    Last Post: November 4th 2009, 09:52 AM
  4. Quick questions on Group Theory - Cosets / Normal Group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 16th 2009, 08:39 AM
  5. Group Theory Question, Dihedral Group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 4th 2008, 10:36 AM

Search Tags


/mathhelpforum @mathhelpforum