# proof that every group of prime order is cyclic

Let $p$ be a prime and $G$ be a group such that $|G|=p$. Then $G$ contains more than one element. Let $g\in G$ such that $g\neq e_{G}$. Then $\langle g\rangle$ contains more than one element. Since $\langle g\rangle\leq G$, by Lagrange’s theorem, $|\langle g\rangle|$ divides $p$. Since $|\langle g\rangle|>1$ and $|\langle g\rangle|$ divides a prime, $|\langle g\rangle|=p=|G|$. Hence, $\langle g\rangle=G$. It follows that $G$ is cyclic.

Title proof that every group of prime order is cyclic ProofThatEveryGroupOfPrimeOrderIsCyclic 2013-03-22 13:30:55 2013-03-22 13:30:55 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Proof msc 20D99 ProofThatGInGImpliesThatLangleGRangleLeG