# Group with prime order?

• Feb 9th 2011, 10:48 PM
CropDuster
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 $p$ contains an element of order $p$.

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.

http://img714.imageshack.us/img714/4...10210at124.png

• Feb 10th 2011, 02:21 AM
tonio
Quote:

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 $p$ contains an element of order $p$.

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.

http://img714.imageshack.us/img714/4...10210at124.png

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