Results concerning Order of Element

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

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

Then the following results apply:

Equal Powers of Finite Order Element

$g^r = g^s \iff k \divides \paren {r - s}$

Element to Power of Remainder

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

Element to Power of Multiple of Order is Identity

$\forall n \in \Z: k \divides n \iff a^n = e$

List of Elements in Finite Cyclic Group

$\set {a^0, a^1, a^2, \ldots, a^{k - 1} }$ is a complete repetition-free list of the elements of $\gen a$