Groups of prime order are cyclic

Let \(|G|=p\) with \(p\) prime, and choose any nonidentity element \(g\in G\). By Lagrange’s theorem, \[ |g|\mid p. \] Since \(g\ne e\), its order is not \(1\), so \[ |g|=p. \] Therefore \[ |\langle g\rangle|=p=|G|, \] and hence \[ G=\langle g\rangle. \] Thus \(G\) is cyclic.