Proving Normal Subgroup Criteria: Exploring the Symmetric and Alternating Groups

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In summary, the conversation discusses a exercise from a book that proves that if N and H are normal subgroups of a group G, then the set NH is also a normal subgroup of G. However, the question arises whether the converse is true, and the speaker struggles to find a counter-example, considering the symmetric group and the alternating group A_{4}. The possibility of N being equal to G is also mentioned.
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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?
 
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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.
 

Related to Proving Normal Subgroup Criteria: Exploring the Symmetric and Alternating Groups

1. What does "NH normal => H normal" mean?

"NH normal => H normal" is a notation used in scientific research to indicate the conversion of an amine group (NH) to a hydrogen atom (H) on a molecule. This process is also known as deprotonation.

2. Why is "NH normal => H normal" important in scientific research?

In many chemical reactions, the presence or absence of the amine group can impact the properties and behavior of a molecule. By converting NH to H, scientists can study the effects of this change on the molecule's structure and function.

3. How is "NH normal => H normal" achieved in the lab?

One common method is through the use of a base, such as sodium hydroxide, which can deprotonate the amine group by accepting a proton. Other methods include using a strong acid or a metal catalyst.

4. Is "NH normal => H normal" reversible?

Yes, the process of deprotonation is reversible. In fact, many chemical reactions involve the protonation and deprotonation of molecules. Researchers can control this process in the lab by adjusting the pH or using specific reagents.

5. What potential applications does "NH normal => H normal" have?

The conversion of NH to H can have significant implications in drug design and development, as well as in the synthesis of new materials. By altering the properties of a molecule through deprotonation, scientists can potentially create more effective and versatile compounds for various purposes.

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