Math 331 - April 21, 1999

Group Actions, Orbits, and Orbit Spaces

Assignment for Friday, April 23

  1. Find the smallest group of isometries of R^{2} that contains the six glide reflections formed using the six ways of composing three different reflections in the sides of a given equilateral triangle. (Note that it is not given that the reflections in the sides are included. Clearly, however, this group must be a subgroup of the smallest group containing the reflections in the three sides.)

  2. Can every isometry of R^{3} be factored as the composition of plane reflections?

  3. What is the orbit space for the action on R^{2} of the subgroup of the translation group consisting of all translations

    (x, y) ---> (x + n, y) n = 0, {+/-} 1, {+/-} 2, ... ?

  4. What are the orbits for the action on R^{3} of the group of its rotations about the origin? And what is the orbit space of R^{3} for this action?


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