Results 1 to 5 of 5

Math Help - Discrete Isometries

  1. #1
    Newbie
    Joined
    Nov 2011
    Posts
    15

    Discrete Isometries

    Let G be a discrete subgroup of Iso(R2). Show that every subgroup of G is also discrete.

    Isn't this true simply because it's a subgroup? So the elements of the subgroup are also in G?
    Last edited by monomoco; November 22nd 2011 at 03:58 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Discrete Isometries

    Quote Originally Posted by monomoco View Post
    Let G be a discrete subgroup of Iso(R2). Show that every subgroup of G is also discrete.

    Isn't this true simply because it's a subgroup? So the elements of the subgroup are also in G?
    I'm confused, if you are talking about the fact that a subgroup of a discrete subgroup is discrete, this really is just the (more general statement) that a subspace of a discrete space is discrete. In particular, you want to show that each element of H\leqslant G is open, but for each h\in H you know (via the fact that G is discrete) that there exists some open set O\subseteq \mathbb{R}^2 such that O\cap G=\{h\}. Clearly though O\cap H=\{h\} and so \{h\} is open in H. Since h was arbitrary the conclusion follows. Make sense?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2011
    Posts
    15

    Re: Discrete Isometries

    We're talking about a subgroup of Iso (R2) so by discrete I have the definition:
    G< Iso (R2) is discrete there exists an E such that

    for all translation t_a in G, a < E
    for all rotations in r_o in G, o<E

    so the motions cannot be arbitrarily small. I understand what you mean, but I don't know how to apply it to this example.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Discrete Isometries

    Quote Originally Posted by monomoco View Post
    We're talking about a subgroup of Iso (R2) so by discrete I have the definition:
    G< Iso (R2) is discrete there exists an E such that

    for all translation t_a in G, a < E
    for all rotations in r_o in G, o<E

    so the motions cannot be arbitrarily small. I understand what you mean, but I don't know how to apply it to this example.
    Oh, well then in that case (the two are equivalent concepts though, which I think you noticed) you must merely note that the same \varepsilon that works for G works for H. So yes, I think it's as simple as you indicated in the initial post.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2011
    Posts
    15

    Re: Discrete Isometries

    I thought I must be missing something. Thanks for your help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Isometries of R2 and R3
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 10th 2012, 06:02 AM
  2. Isometries of R2
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 4th 2012, 08:17 PM
  3. Isometries of R3
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 4th 2012, 03:38 AM
  4. isometries of H
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 24th 2010, 02:53 PM
  5. Isometries
    Posted in the Geometry Forum
    Replies: 1
    Last Post: December 7th 2009, 10:52 PM

Search Tags


/mathhelpforum @mathhelpforum