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Math Help - subgroup question

  1. #1
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    subgroup question

    Show that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28.
    Here is what I have so far:
    Let H be a normal subgroup of order 4. Then |G/H|=42=2*3*7, so then G?N has a unique, and therefore normal Sylow 7-subgroup, lets call it K.
    I was told to use the correspondence theorem, but I am not sure where it works in here. any ideas?
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  2. #2
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    Quote Originally Posted by wutang View Post
    Show that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28.
    Here is what I have so far:
    Let H be a normal subgroup of order 4. Then |G/H|=42=2*3*7, so then G?N has a unique, and therefore normal Sylow 7-subgroup, lets call it K.
    I was told to use the correspondence theorem, but I am not sure where it works in here. any ideas?


    So G/H has a unique, normal sugroup K/H of order 7, which then pulls back under the canonical projection (this is the correspondence theorem) to

    a subgroup K\leq G of order 7\cdot 4=28 (why?!? Look at the respective indexes which, again by the CT, stay the same...) which, again by the corr. theorem, is

    also normal in G

    Tonio
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