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Math Help - Subgroup of factor groups

  1. #1
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    Subgroup of factor groups

    Let G be a group, and let H be normal in G. Prove that every subgroups of G/H is of the form K/H, where K is a subgroup of G with H \subset K.

    My proof so far:

    Let Y:G \rightarrow \frac{G}{H} , let X be a subgroup of G/H. By a theorem, Y^{-1}(X) = \{t \in \frac{G}{H} : Y(t) \in X \} is a subgroup of G.

    Any help from here?

    Thanks.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let G be a group, and let H be normal in G. Prove that every subgroups of G/H is of the form K/H, where K is a subgroup of G with H \subset K.

    My proof so far:

    Let Y:G \rightarrow \frac{G}{H} , let X be a subgroup of G/H. By a theorem, Y^{-1}(X) = \{t \in \frac{G}{H} : Y(t) \in X \} is a subgroup of G.

    Any help from here?

    Thanks.
    I have no idea what it means "of the form K/H"
    Given the canonical homormorphism \gamma: G\mapsto G/H.
    If L\leq G/H is a subgroup, i.e. L = qH then,
    \gamma^{-1} [L] = \{ x\in G | \gamma (x) \in qH \} = \{ x\in G | xH \in qH \} = qH .
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