prove that if the order of group G is a prime number then G is cyclic.

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- Oct 8th 2008, 12:17 PMmandy123prove
prove that if the order of group G is a prime number then G is cyclic.

- Oct 8th 2008, 12:44 PMThePerfectHacker
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 $\displaystyle \left< a \right> = G$.