Abstract Algebra: Theory and Applications · Problem 6.5.14

The order of $g$ is the smallest positive integer $m$ such that $g^m = e$—it measures how many times you must apply $g$ before returning to the identity. The key insight is the division algorithm: if $g^n = e$, you can write $n = mq + r$ where $m$ is the order and $0 \le r < m$. Then $g^r = g^{n-mq} = (g^n)(g^m)^{-q} = e$, which forces $r = 0$ by minimality of $m$. Therefore $m$ divides $n$.

The order of $g$ is the smallest positive integer $m$ such that $g^m = e$—it measures how many times you must apply $g$ before returning to the identity. The key insight is the division algorithm: if $g^n = e$, you can write $n = mq + r$ where $m$ is the order and $0 \le r < m$. Then $g^r = g^{n-mq} = (g^n)(g^m)^{-q} = e$, which forces $r = 0$ by minimality of $m$. Therefore $m$ divides $n$.