Question

Let G be a finite group with more than 1 element. Show that G has an element of Prime order.

Answer 1

Suppose \(G\) is a group of finite order greater than \(1\). Then \(G\) has a non trivial element \(x\in G\). Suppose \(x\) has order prime, then we are done. Suppose \(x\) has non prime order \(k\). Then \(k=p*k_0\) where \(p,k_o\in\mathbb{N}\) and \(p\) is prime. So \(x^k=e=x^{pk_0}=(x^{k_0})^p\). Now \(|x^{k_0}|=p\) and we re done.