Answer to Question #138326 in Abstract Algebra for J

Question

Prove that if (G,*) is a finite group, and g is an element of G, then there exists a positive integer n such that g^n =e

Answer 1

Consider the sequence of powers of an element $g\in G$:

$g, g^2, g^3,\dots, g^k,\dots$

Since $G$ is finite and $G$ contains this sequence, there exist $t, s\in\mathbb N, t<s$, such that $g^s=g^t$. Multiply both part of this equality by $(g^t)^{-1}=g^{-t}$. Then $g^{s-t}=e$ and $s-t>0$. Let $n=s-t\in\mathbb N$. We conclude that $g^n=e.$