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A lemma in proving Sylow's theorem

  1. Apr 12, 2015 #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
     
  2. jcsd
  3. Apr 13, 2015 #2

    micromass

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