Proof of Sylow: Let G be a Finite Group, H and K Subgroups of G

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In summary, the proof of Sylow's theorem states that for any finite group G, if H and K are subgroups of G such that the order of H is a power of a prime p and the order of K is a power of a different prime q, then the intersection of H and K must be the identity element. This implies that H and K are disjoint subgroups, and the product of their orders is equal to the order of G. This result is important in the study of finite groups and has many applications in group theory.
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Let G be a finite group, H and K subgroups of G such that G=HK. Show that there exists a p-Sylow subgroup P of G such that P=(P∩H)(P∩K).

I found this proof and it is clear http://math.stackexchange.com/questions/42495/sylow-subgroups but I do not understand step 3 which is "It is clear in this situation that P=(P∩H)(P∩K)". Help.
 
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moont14263 said:
Let G be a finite group, H and K subgroups of G such that G=HK. Show that there exists a p-Sylow subgroup P of G such that P=(P∩H)(P∩K).

I found this proof and it is clear http://math.stackexchange.com/questions/42495/sylow-subgroups but I do not understand step 3 which is "It is clear in this situation that P=(P∩H)(P∩K)". Help.

+

Well, after part (3) in the stackexchange there's a further poster who also had a problem with this step, and I think it isn't THAT clear, as the other guy wrote, that P = (P/\H)(P/\K)...
The same question as that further poster asked came to my mind: is normality something we can do without?

It'd be interesting to know where did you find this problem.

DonAntonio
 
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It is in Finite Group Theory I. Martin Isaacs page 18
 

1. What is the proof of Sylow's theorem?

The proof of Sylow's theorem states that for any finite group G, if H and K are subgroups of G with orders that are relatively prime, then there exists a subgroup J of G such that J is isomorphic to the direct product of H and K. In other words, there exists a subgroup of G that contains both H and K as subgroups, and the order of this subgroup is the product of the orders of H and K.

2. Why is Sylow's theorem important in group theory?

Sylow's theorem is important in group theory because it provides a way to analyze the structure of a finite group by breaking it down into smaller, more manageable subgroups. It also allows for the classification of finite groups into different categories based on their subgroups, which can help in understanding the properties and behaviors of these groups.

3. What is the significance of the relatively prime condition in Sylow's theorem?

The relatively prime condition in Sylow's theorem is significant because it guarantees that the subgroup J that is isomorphic to the direct product of H and K is a proper subgroup of G. This is important because it ensures that the subgroup is smaller than the original group and therefore provides useful information about the structure of G.

4. Can Sylow's theorem be extended to infinite groups?

No, Sylow's theorem only applies to finite groups. For infinite groups, there are other theorems and techniques that can be used to analyze their structure and subgroups.

5. How is Sylow's theorem used in other areas of mathematics?

Sylow's theorem has applications in other areas of mathematics, such as algebraic geometry and number theory. It can also be used in cryptography and coding theory to analyze the structure of certain codes and ciphers. Additionally, Sylow's theorem has connections to other theorems and concepts in group theory, making it an important tool for understanding the properties of groups in general.

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