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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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This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.
 Spring '08
 Staff
 Math, Algebra

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