# prove

• October 8th 2008, 12:17 PM
mandy123
prove
prove that if the order of group G is a prime number then G is cyclic.
• October 8th 2008, 12:44 PM
ThePerfectHacker
Quote:

Originally Posted by mandy123
prove that if the order of group G is a prime number then G is cyclic.

Let $a\in G - \{ e \}$ then $\text{ord}(a) > 1$ and $\text{ord}(a)$ divides $p$ by Lagrange's theorem.
Therefore, $\text{ord}(a) = p$ since $p$ is a prime. Therefore $\left< a \right> = G$.