Proving Normality of [G,G] in G: A Commutator Question

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In summary, the conversation is about proving that the group [G,G], consisting of elements of the form ghg^-1h^-1 where g,h are in G, is normal in G. The person tried writing it in a certain way but was not successful and is looking for hints. The suggested hint is to conjugate two elements g and h by the same element and compute the commutator.
  • #1
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Perhaps someone will help me in this.
I need to prove that the group [G,G] of elements of the form [tex]gh g^{-1}h^{-1}[/tex] where g,h in G, is normal in G, i.e if k is in G, then [tex]kghg^{-1}h^{-1}k^{-1}=aba^{-1}b^{-1}[/tex] for some a,b in G.
I tried writing it as [tex]kghkk^{-1}g^{-1}h^{-1}k^{-1}[/tex], but here is where my attmempt run down the mill, as in not successful.
Any hints?
 
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  • #2
Conjugate two elements g,h by the same element and compute the commutator.
 
  • #3
Ok, thanks.
 

FAQ: Proving Normality of [G,G] in G: A Commutator Question

What is the meaning of "Proving Normality of [G,G] in G: A Commutator Question"?

This question refers to a mathematical concept in group theory, specifically in the context of commutators. To prove that [G,G] (the commutator subgroup of G) is normal in G means to show that it satisfies certain properties, namely that for any element in G, its commutator with any other element will also be in G.

Why is proving normality of [G,G] in G important?

Proving normality is important because it allows for easier manipulation and analysis of a group. If [G,G] is normal in G, then it can be used to simplify complex group operations and can also provide insight into the structure of the group.

What are some common methods for proving normality of [G,G] in G?

One common method is to show that [G,G] is a subset of the center of G (the set of elements that commute with all other elements in the group). Another method is to use the properties of the commutator subgroup, such as its closure under conjugation and taking inverses.

Are there any real-world applications of proving normality of [G,G] in G?

Yes, the concept of normality in group theory has applications in many areas of mathematics, physics, and computer science. For example, in cryptography, normal subgroups play a crucial role in the security of certain encryption algorithms.

Is proving normality of [G,G] in G a difficult task?

It depends on the specific group and the methods used. In some cases, it may be relatively straightforward to prove normality, while in others it may be a more challenging task. It also requires a solid understanding of group theory and its properties.

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