proof that every group of prime order is cyclic

The following is a 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
Canonical name	ProofThatEveryGroupOfPrimeOrderIsCyclic
Date of creation	2013-03-22 13:30:55
Last modified on	2013-03-22 13:30:55
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	7
Author	Wkbj79 (1863)
Entry type	Proof
Classification	msc 20D99
Related topic	ProofThatGInGImpliesThatLangleGRangleLeG