A lemma in proving Sylow's theorem

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In summary, the conversation is about proving a fact in Sylow's first theorem, which states that in a finite group G, if K is a Sylow p-subgroup of G, then any element x in the normalizer of K with order p must also be in K. The conversation ends with a request for tips on proving this fact for a general finite group, as it has already been proven for Abelian groups. The idea suggested is to show that the group generated by K and {x} is a p-group that contains K.
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Bipolarity
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I'm following the theorems/proofs of Contemporary Abstract Algebra by Gallian, 8th edition, and in proving Sylow's first theorem, the text assumes the following fact, which I am unsure how to prove, and was looking for tips:

Let G be a finite group and let K be a Sylow p-subgroup of G of order ##p^{k}##.
Let ##x## be an element in ##N(K)## and suppose that ##|x| = p##. Prove that ##x \in K##.

Any ideas?
I have been able to prove it for the Abelian groups (it's trivial then), but for a general finite group?

Thanks!

BiP
 
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  • #2
Try to show that the group generated by ##K## and ##\{x\}## is a ##p##-group that contains ##K##.
 

What is a lemma in proving Sylow's theorem?

A lemma is a smaller, simpler version of a theorem that is used to prove a larger, more complex theorem. In the context of Sylow's theorem, a lemma is a preliminary result that helps to establish the main theorem.

Why is a lemma necessary in proving Sylow's theorem?

A lemma is necessary because Sylow's theorem is a complex and multifaceted theorem that requires multiple steps to prove. By breaking it down into smaller, more manageable lemmas, it becomes easier to understand and prove the main theorem.

What is the main role of the lemma in proving Sylow's theorem?

The main role of the lemma in proving Sylow's theorem is to provide a key step in the overall proof. Often, lemmas serve as building blocks that, when combined, lead to the proof of the main theorem. In Sylow's theorem, the lemma helps to establish the existence of Sylow subgroups.

How does the lemma contribute to the overall proof of Sylow's theorem?

The lemma contributes to the proof of Sylow's theorem by providing a crucial piece of evidence that supports the main theorem. Without the lemma, it would be much more difficult to prove Sylow's theorem, as it serves as a bridge between the initial assumptions and the final conclusion.

Are there different lemmas that can be used to prove Sylow's theorem?

Yes, there are multiple lemmas that can be used to prove Sylow's theorem. Different mathematicians may have their own preferred lemmas or may use different approaches to prove the main theorem. However, all of these lemmas ultimately contribute to the overall proof of Sylow's theorem.

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