Prove that if $G=\langle x\rangle$ for every $x\in G\setminus \{1_G\}$, then $|G|$ is prime

Question

I am trying to prove that the following statements are equivalent:

(i) the group $G$ is generated by every element of the group, except the neutral element

(ii) $G$ is trivial or $|G|$ is prime

I used Lagrange to go from (ii) to (i), but I am having trouble proving that (i) implies (ii). If $G$ is cyclic for every $x$ in $G$, why does the order have to be a prime number?

Answer 1

Suppose $G$ has non-prime order, so $|G|=mn$ for $m, n>1$, and let $G=\langle x\rangle$.

Then can $\langle x^m\rangle=G$?