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Math Help - Order of the factor group G/N

  1. #1
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    Order of the factor group G/N

    Question: Suppose G is a group with |G| = 30 and N is a normal subgroup of G. WHat are the possible orders for the group G/N

    I know by Lagrange's theorem that |N| divides |G| so |N| = {1,2,3,5,10,15,30}.

    I also know that |G/N| / |N| = |G| / |N| so I believe |G/N| = |N| = {1,2,3,5,10,15,30}

    Am I off base or is there a better way to approach this?

    Thanks
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  2. #2
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    Quote Originally Posted by temp31415 View Post
    Question: Suppose G is a group with |G| = 30 and N is a normal subgroup of G. WHat are the possible orders for the group G/N

    I know by Lagrange's theorem that |N| divides |G| so |N| = {1,2,3,5,10,15,30}.

    I also know that |G/N| / |N| = |G| / |N| so I believe |G/N| = |N| = {1,2,3,5,10,15,30}

    Am I off base or is there a better way to approach this?

    Thanks
    Thus, the possible normal subgroups that |N| can be are: 1,2,3,5,6,10,15,30
    Thus, |G/N|=|G|/|N| are: 1,2,3,5,6,10,15,30

    Just note this might not be the best answer. Meaning using some additional results we might be able to disregard which orders are possible and impossible. But that is a more difficult task.
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