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Math Help - abstract algebra help

  1. #1
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    abstract algebra help

    Given K is a normal subgroup of a group G. The map \psi : G --> G/K by \psi(g) = \phi(g)K for g \in G is a homomorphism and \phi \in Aut(G).
    Show \psi(K) = (\phi(K)K)/K.
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  2. #2
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    Since K is a subgroup of G, \psi(K) is a subgroup of G/K. So for every x \in \psi(K), x \in G/K, and then x = gK for some g \in G.
    I need to show that \psi(K) and (\phi(K)K)/K are subsets of each other, but I'm stuck in here, did I start this right?
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