Element to Power of Remainder

Theorem

Let $G$ be a group whose identity is $e$.

Let $a \in G$ have finite order such that $\order a = k$.

Then:

$\forall n \in \Z: n = q k + r: 0 \le r < k \iff a^n = a^r$

Let $n \in \Z$.

We have:

$\ds n$

$=$

$\ds q k + r$

$\ds \leadstoandfrom \ \ $

$\ds n - r$

$=$

$\ds q k$

$\ds \leadstoandfrom \ \ $

$\ds k$

$\divides$

$\ds \paren {n - r}$

$\blacksquare$

1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 38.4$ Period of an element: $\text{(i)}$