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Math Help - well-defined proof

  1. #1
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    well-defined proof

    A, A^{*}, B, B^{*} be four subgroups of a group G with A \vartriangleleft A^{*} and B \vartriangleleft B^{*}.
    Let D=(A^{*} \cap B)(A \cap B^{*}).
    I have shown the following:
    (1) B \vartriangleleft B(A^{*} \cap B^{*})
    (2) (A^{*} \cap B) \vartriangleleft (A^{*} \cap B^{*})
    (3) (A \cap B^{*}) \vartriangleleft (A^{*} \cap B^{*})
    (4) D \vartriangleleft (A^{*} \cap B^{*})

    If x \in B(A^{*} \cap B^{*}), then x=bc for b \in B and c\in (A^{*} \cap B^{*});
    define f:B(A^{*} \cap B^{*}) \rightarrow (A^{*} \cap B^{*})/D by f(x)=Dc.

    I'm facing problem to show that f is well-defined.
    I started with x_{1}=x_{2};
    implies that b_{1}c_{1}=b_{2}c_{2}. ----------( *)
    I'm stuck here.
    I know I need to get c_{1}c_{2}^{-1} \in D; ------------( **)
    implies that Dc_{1}=Dc_{2};
    finally  f(x_{1})=f(x_{2}).

    How to continue from ( *) to ( **)?
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  2. #2
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    I have another question.

    How to show that Ker\:f=B(A \cap B^{*}) ?
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