Results 1 to 5 of 5

Math Help - Geometric example of a group

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    54

    Geometric example of a group

    Hi,

    I need to come up with a geometric example of a group. I've been given an example, the set of all symmetries of an equilateral triangle. But I then have to show that the axioms are satisfied and I dont really understand how they are.

    I know the axioms are the following, given a set X and an operation *: (X x X) --> X then (X,*) is a group if:
    (1) there exists a neutral element e such that x*e=e*x=x for all x in X
    (2) the operation is associative
    (3) every element x in X has an inverse y such that x*y=y*x=e

    Please help,

    Katy
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Amer's Avatar
    Joined
    May 2009
    From
    Jordan
    Posts
    1,093
    Quote Originally Posted by harkapobi View Post
    Hi,

    I need to come up with a geometric example of a group. I've been given an example, the set of all symmetries of an equilateral triangle. But I then have to show that the axioms are satisfied and I dont really understand how they are.

    I know the axioms are the following, given a set X and an operation *: (X x X) --> X then (X,*) is a group if:
    (1) there exists a neutral element e such that x*e=e*x=x for all x in X
    (2) the operation is associative
    (3) every element x in X has an inverse y such that x*y=y*x=e

    Please help,

    Katy
    what is the operation, you said the set is all symmetries of an equilateral triangle, what is operation defined on this set ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    The set of all symmtries of an equilateral triangle contains
    six elements : three line symmtries, and three rotational symmtries;
    It is actually the collection of all the transformation which keep the equilateral triangle remain unchange.
    It is easy to verify that the three axioms of Group is satisfied.
    The operation is the composition of transform(mapping).
    Last edited by Shanks; November 27th 2009 at 04:56 AM. Reason: add something
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2008
    Posts
    54
    ahh, that makes a little more sense now. Thank you

    So is the inverse to an element just the element itself, seeing as none of the transformations change the triangle? The neutral element the original triangle? How do you prove that it is associative?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    the group G=\{I,R,R^2,T,TR,TR^2\}where I is the identity transform, R is the Rotational Transformation of 120 degree with respect to the center of the equilateral triangle, T is the symmetrical transformation with respect to one(fixed) of the three symmetry line, and R^3=I, RT=TR^2.
    So,for example,the inverse of R is [tex]R^2[tex], the inverse of T is T itself.
    the composition of transformation obviously satisfies the associate law.
    It is very easy to virefy that all axioms of group are satisfied, you can do this by write down the operation table of G, then you can understand the group better !
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: May 23rd 2011, 02:36 AM
  2. Question about Asymptotic and Geometric Group theory
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 8th 2011, 01:03 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. Geometric Progression or Geometric Series
    Posted in the Math Topics Forum
    Replies: 8
    Last Post: October 8th 2009, 07:31 AM

Search Tags


/mathhelpforum @mathhelpforum