Proof that finite group contains an element of prime order I tried to prove the following claim but it seemed a bit too easy:

Let $G$ be a finite group. Then $G$ contains an element of prime order.

Please could someone tell me if my proof is correct or if I'm missing something?
Let $g$ be any non-identity element of $G$. Let $p$ be any prime factor of $|g|$. Then $g^{|g|\over p}$ has prime order $p$. $\Box$

No the proof is not correct. It is possible that $g^{\frac{|g|}{p}}=e$. The theorem you are stating is commonly called Cauchy's theorem.  There is an elementary proof here using the class equation. 
