# A lemma in proving Sylow's theorem

1. Apr 12, 2015

### Bipolarity

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. Apr 13, 2015

### micromass

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