Is NH Always a Normal Subgroup of G When N and H Are Normal Subgroups?

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SUMMARY

The discussion centers on the properties of normal subgroups within group theory, specifically whether the product of two normal subgroups, NH, is always a normal subgroup of G when both N and H are normal subgroups of G. The user explores the possibility of a counter-example using the symmetric group, particularly the alternating group A4, and considers the Klein four-group as a potential non-normal subgroup. The conclusion drawn is that while NH is a normal subgroup when both N and H are normal, the converse does not hold universally, as demonstrated through specific subgroup constructions.

PREREQUISITES
  • Understanding of group theory concepts, specifically normal subgroups.
  • Familiarity with the symmetric group and its properties.
  • Knowledge of the alternating group A4 and its subgroup structure.
  • Basic understanding of the Klein four-group and its characteristics.
NEXT STEPS
  • Research the properties of normal subgroups in group theory.
  • Study the structure and properties of the symmetric group Sn.
  • Examine the subgroup lattice of the alternating group A4.
  • Explore counter-examples in group theory involving normal and non-normal subgroups.
USEFUL FOR

Mathematicians, particularly those specializing in abstract algebra, students studying group theory, and anyone interested in the properties of normal subgroups and their applications in group structures.

Bleys
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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?
 
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What if N=G?
 
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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