problem on group theory

Question

Prove that a group with more than one element contains an element of prime order.

Answer 1

So your group has a non identity element $a$. Since the group is finite, the order of $a$ is finite. Say $|a| = p_1^{n_1}\dotsb p_k^{n_k}$ where the $p_i$ are prime. What is the order of the element $$\large a^{\;p_1^{n_1}\dotsb p_k^{n_k-1}}\;?$$

Answer 2

pick a non identity element $g$, suppose $p$ divides $|g|$, then $g^{|g|/p}$ has order $p$.