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Math Help - Group actions

  1. #1
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    Group actions

    Hey, We can define a group action in two ways:

    1)
    A group G is said to act on a set X, if there exists a group homomorphism ψ : G --> S(X) [the symmetric group of X].
    2)Equivalently, a group G acts on a set X, if there is a map from G x X --> X which assigns to each ordered pair <g,x>--->g.x, such that :
    For all x Є X, e.x = x
    (h.g).(x) = h.(g.x), where h, g Є G, and x Є X
    How are these two definitions equivalent?
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    (1)\ \implies\ (2)

    Consider the map \phi:G\times X\to X,\ \phi((g,x))=\psi(g)(x).


    (2)\ \implies\ (1)

    For each g\in G, define \psi_g:X\to X by \psi_g(x)=g\cdot x for all x\in X. Then \psi_g\in S(X) for all g\in G (\psi_g is a bijection from X to itself) and the required homomorphism \psi:G\to S(X) is defined by \psi(g)=\psi_g for all g\in G (\psi_{hg}=\psi_h\psi_g for all h,g\in G).
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