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Group Action and a Cartesian Product

  1. Sep 1, 2009 #1
    Suppose a group G and it acts on a set X and a set Y.

    (a) A simple group action on the cartesian product would be defined as such:

    G x (X x Y) --> (X x Y)

    to prove this is a group action could I just do this:

    Suppose a g1 and g2 in G. g1*(g2*(x,y))=g1*g2(x). This is obvious. Basically is the proof extremely easy. I just grabbed this example out of a book and was wondering if I am close.
  2. jcsd
  3. Sep 2, 2009 #2
    Not sure what your question is but you seem to have a typo in your last equation.

    g.(x,y) = (g.x,g.y) works on the direct product. Verify this directly.
  4. Sep 2, 2009 #3
    I guess the question is asking for me to define and prove a very general and simple action on that cartesian product.
  5. Sep 3, 2009 #4
    Yeah, basically you just need to show that g1*(g2*(x))=(g1*g2)(x) and g1*(g2*(y))=(g1*g2)(y), which is really what the definition states. I think.
  6. Sep 3, 2009 #5
    For the sake of group theory, when g operates on y in Y, would it need to be using a different operator such as '$'
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