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Group generated by a nonzero translation

  1. Apr 8, 2009 #1
    Let G1 be the group generated by a nonzero translation and G2 be the group generated by a glide reflection. Show that G1 and G2 are isomorphic.

    Here is how I started:

    G1 = <Tb> where b[tex]\in[/tex] C and Tb(z) = z+b

    G2 = <ML [tex]\circ[/tex] Tc> where c and L are parallel to each other.

    Let's define a function [tex]\Phi[/tex]: G1 [tex]\rightarrow[/tex] G2

    Then if [tex]\Phi[/tex] is a homomorphism and a bijection, it is an isomorphism.

    But here lies my problem, I do not know what to make [tex]\Phi[/tex] equal to. Maybe this isn't the right way of approaching this problem.

    Thanks in advance.
     
  2. jcsd
  3. Apr 8, 2009 #2
    Re: Isomorphisms

    How can you perform a translation using a glide reflection? How can you perform a glide reflection with a translation?
     
  4. Apr 8, 2009 #3
    Re: Isomorphisms

    A glide reflection composed with itself is a translation. And you get a glide reflection by composing a reflection with a translation.
     
  5. Apr 8, 2009 #4
    Re: Isomorphisms

    So informally, translation = 2 * glide reflection and glide reflection = reflection + translation. Since you need a reflection to get a glide translation, I don't see any way of getting an isomorphism between the two groups.
     
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