prove that if the order of group G is a prime number then G is cyclic.
Let $\displaystyle a\in G - \{ e \}$ then $\displaystyle \text{ord}(a) > 1$ and $\displaystyle \text{ord}(a)$ divides $\displaystyle p$ by Lagrange's theorem.
Therefore, $\displaystyle \text{ord}(a) = p$ since $\displaystyle p$ is a prime. Therefore $\displaystyle \left< a \right> = G$.