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NH normal => H normal

  1. Jul 12, 2010 #1
    There was an exercise in a book to prove that given N is a normal subgroup of a group G if H is also another normal subgroup of G the NH (the set of elements of the form nh for n in N and h in H) is a normal subgroup of G. That was all fine but I was wondering if the converse is true. Considering the exercise didn't ask to do this I'm guessing no, but I'm finding it hard to create a counter-example.
    I'm trying to use the symmetric group. I thought that the alternating group [itex]A_{4}[/itex] could maybe be constructed from the Klein 4 group and another non-normal subgroup but I don't know how to show this, or if it's even true. Are there maybe simpler examples?
  2. jcsd
  3. Jul 12, 2010 #2


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    What if N=G?
  4. Jul 12, 2010 #3
    If [itex]H[/itex] is the klein-4 group, and [itex]N = <g>[/itex] where [itex]g\in A4\setminus H[/itex], then the product [itex]NH=A4[/itex] just based on index.
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