A lemma in proving Sylow's theorem

  • Thread starter Bipolarity
  • Start date
  • #1
775
1
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
 

Answers and Replies

  • #2
22,089
3,292
Try to show that the group generated by ##K## and ##\{x\}## is a ##p##-group that contains ##K##.
 

Related Threads on A lemma in proving Sylow's theorem

Replies
1
Views
1K
  • Last Post
Replies
5
Views
674
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
6
Views
2K
Replies
1
Views
3K
  • Last Post
Replies
1
Views
609
Replies
2
Views
2K
Top