Unformatted text preview: N ± G . Prove that  G / HN  and  N / N ∩ H  divide  G / H  . Note that HN ≤ G , because N ± G (we need one of the subgroups in HN to be normal for HN to be a subgroup). Then  G / HN  =  G  /  HN  = (  G  /  H  ) / (  HN  /  H  ) =  G / H  /  HN / H  . Since  HN / H  is a positive integer, we deduce that  G / HN  divides  G / H  . By one of the isomorphism theorems, we have HN / N ∼ = H / H ∩ N and hence  HN  /  N  =  H  /  H ∩ N  . Therefore  N / N ∩ H  =  N  /  N ∩ H  =  HN  /  H  = (  G  /  H  ) /  G  /  HN  =  G / H  /  G / HN  . Since  G / HN  is a positive integer, we conclude that  N / N ∩ H  divides  G / H  ....
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 Spring '08
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 Math, Algebra, Prime number, Cyclic group, N /N

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