Element to the Power of Remainder

Theorem

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

Let $a \in G$ have finite order such that $\left|{a}\right| = k$.

Then:

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

Proof

Let $n \in \Z$.

We have:

$n = q k + r \iff n - r = q k \iff k \backslash \left({n - r}\right)$

The result follows from Equal Powers of Finite Order Element.

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 41$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 38.4 \ \text{(i)}$