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  1. #1
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    Transitive

    Prove that the notion of group isomorphism is transitive.

    My proof so far:

    Suppose G, H, and K are groups.

    Let the function f be the isomorphism from G onto H, then f(g_{1}g_{2})=f(h_{1})f(h_{2}) with f is onto and 1-1.

    Now let the function t be the isomorphism from H onto K, then t(h_{1}h_{2})=t(k_{1})t(k_{2}) with t is onto and 1-1.

    My plan is to get a function, say X, such that X(g_{1}g_{2}) = X(k_{1}k_{2})

    First I tried with X = f(t), but it doesn't really work out, so am I at least doing something right here?

    Thanks.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Prove that the notion of group isomorphism is transitive.

    My proof so far:

    Suppose G, H, and K are groups.

    Let the function f be the isomorphism from G onto H, then f(g_{1}g_{2})=f(h_{1})f(h_{2}) with f is onto and 1-1.
    That equation should be f(g_{1}g_{2})=f(g_{1})f(g_{2}).

    Now let the function t be the isomorphism from H onto K, then t(h_{1}h_{2})=t(k_{1})t(k_{2}) with t is onto and 1-1.
    That equation should be t(h_{1}h_{2})=t(h_{1})t(h_{2}).

    My plan is to get a function, say X, such that X(g_{1}g_{2}) = X(k_{1}k_{2})
    That equation should be X(g_{1}g_{2}) = X(g_{1})X(g_{2}).

    First I tried with X = f(t), but it doesn't really work out, so am I at least doing something right here?
    The reason it doesn't "really work out" may be that you are trying to combine f and t in the wrong order. If you define X(g) = t(f(g)) then X(g_1g_2) = t(f(g_1g_2)) = t(f(g_1)f(g_2)) = t(f(g_1))t(f(g_2)) = X(g_1)X(g_2). Of course, you also have to verify that X is onto and 1-1.
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  3. #3
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    Quote Originally Posted by tttcomrader View Post
    Prove that the notion of group isomorphism is transitive.

    My proof so far:

    Suppose G, H, and K are groups.

    Let the function f be the isomorphism from G onto H, then f(g_{1}g_{2})=f(h_{1})f(h_{2}) with f is onto and 1-1.

    Now let the function t be the isomorphism from H onto K, then t(h_{1}h_{2})=t(k_{1})t(k_{2}) with t is onto and 1-1.

    My plan is to get a function, say X, such that X(g_{1}g_{2}) = X(k_{1}k_{2})

    First I tried with X = f(t), but it doesn't really work out, so am I at least doing something right here?

    Thanks.
    If \phi: G_1\mapsto G_2 is an isomorphism and \psi: G_2\mapsto G_3 is an isomorphism then \psi \circ \phi:G_1\mapsto G_3 is an isomorphism.
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