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Math Help - Normal subgroup question

  1. #1
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    Normal subgroup question

    Hi, everyone! This is my first post on the forums.

    There is a practice problem I've found that states:

    "Given N1, a normal subgroup of G1, and N2, a normal subgroup of G2, show N1 X N2 is a normal subgroup of G1 X G2."

    Now I've been doing a lot of approaches, but the one that seems to be working out for me hinges on if I can assume that N1 is a normal subgroup of G1 X G2 (which then means I can assume as well that N2 is a normal subgroup of G1 X G2). But can I make this assumption? Is it true? Why or why not?

    EDIT: I realize that I should be working with N1 X Z1 and N2 X Z1 above so that they can be normal subgroups (which are isomorphic to N1 and N2 respectively). I'm still unsure of how to show that it is the case that N1 X Z1 is a normal subgroup of G1 X G2 and N2 X Z1 is a normal subgroup of G1 X G2.

    Thanks for any input!
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  2. #2
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    Elements of G_1 \times G_2 are pairs of elements of G1 and G2, while elements of N_1 \times N_2 are pairs of elements of N1 and N2.
    If g \in G_1 \times G_2, and h \in N_1 \times N_2 how can you write them in terms of each group? How is group multiplication defined in G_1 \times G_2? What can you then say about g^{-1}hg?
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  3. #3
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    Here was my outline for doing the whole problem:

    N_1 \times N_2 \triangleright G_1 \times G_2 \leftrightarrow (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2

    So, to show that N_1 \times N_2 \triangleright G_1 \times G_2, I want to show that (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2

    [If I had N_1 \times Z_1 \triangleright G_1 \times G_2 and N_1 \times Z_1 \cong N_1, then (g_1g_2)(n_1)(g_1g_2)^{-1} \in N_1]

    Similarly,

    [If I had N_2 \times Z_1 \triangleright G_1 \times G_2 and N_2 \times Z_1 \cong N_2, then (g_1g_2)(n_2)(g_1g_2)^{-1} \in N_2]

    Then by looking at N_1 \times N_2, I would multiply the previous results to obtain this:

    (g_1g_2)(n_1)(g_1g_2)^{-1}(g_1g_2)(n_2)(g_1g_2)^{-1} \in N_1 \times N_2

    = (g_1g_2)(n_1)g_2^{-1}g_1^{-1}g_1g_2(n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2

    = (g_1g_2)(n_1)g_2^{-1}g_2(n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2

    = (g_1g_2)(n_1n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2

    = (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2 QED.

    Is this an acceptable approach? The material in the colored brackets is what I'm not comfortable assuming.
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