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Math Help - Conjugate Stabilisers

  1. #1
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    Conjugate Stabilisers

    How do I go about starting to show that, if G acts on X and x is in X, with y=g(x) for some g in G, then the stabiliser of y is the conjugate of the stabiliser of x?
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  2. #2
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    you know that y = g(x). it follows that g^-1(y) = x.

    now suppose that a is in Stab(x).

    then gag^-1(y) = g(a(g^-1(y))) = g(a(x)) = g(x) = y.

    thus gag^-1 is in Stab(y).

    similarly, if b is in Stab(y), then g^-1bg is in Stab(x), so

    b = g(g^-1bg)g^-1 is in g(Stab(x))g^-1.

    from this we can conclude that g(Stab(x))g^-1 = Stab(y).

    (we only need the first part if G is finite, since then

    g(Stab(x))g^-1 contained in Stab(y) shows they are equal).
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