TITLE: Show that every group of prime order is cyclic
QUESTION [37 upvotes]: Show that every group of prime order is cyclic.

I was given this problem for homework and I am not sure where to start. I know a solution using Lagrange's theorem, but we have not proven Lagrange's theorem yet, actually our teacher hasn't even mentioned it, so I am guessing there must be another solution. The only thing I could think of was showing that a group of prime order $p$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. Would this work?  
Any guidance would be appreciated.

REPLY [15 votes]: Let $G$ be a group of prime order $p > 1$. Let $a \in G$ such that $a \neq e$. Note that $\langle a \rangle \leq G$, so by Lagrange's theorem $|\langle a \rangle|$ divides $|G|$. Since $p$ is prime, either $|\langle a \rangle| = 1$ or $|\langle a \rangle| = p$, but $|\langle a \rangle| = 1$ is not possible since that would imply $\langle a \rangle = \{e\}$ and therefore $a = e$. So $|G| = p = |\langle a \rangle|$, and therefore $G = \langle a \rangle$.