# Thread: Group with prime order?

1. ## Group with prime order?

Hi guys,

I've been asked to prove the following:
Prove that every group whose order is a power of a prime $\displaystyle p$ contains an element of order $\displaystyle p$.

This is the proof I've come up with. I feel like it is really long and contrived. The only tools that I have at my disposal is LaGrange's Theorem and the associated corollaries.

2. Originally Posted by CropDuster
Hi guys,

I've been asked to prove the following:
Prove that every group whose order is a power of a prime $\displaystyle p$ contains an element of order $\displaystyle p$.

This is the proof I've come up with. I feel like it is really long and contrived. The only tools that I have at my disposal is LaGrange's Theorem and the associated corollaries.

I think the above is too long. Immediately after "assume $\displaystyle |<x>|\neq p$ " you can finish swiftly: then
$\displaystyle |<x>|=p^m\Longrightarrow$ which means $\displaystyle ord(x)=p^m\Longrightarrow ord(x^{p^{m-1}})=p$ and we're done.