Results concerning Order of Element

Theorems

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

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

Then the following results apply:

Equal Powers of Finite Order Element

$g^r = g^s \iff k \backslash \left({r - s}\right)$.

Element to the Power of Remainder

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

Element to the Power of Multiple of Order

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

List of Elements in Finite Cyclic Group

$\left\{{a^0, a^1, a^2, \ldots, a^{k - 1}}\right\}$ is a complete repetition-free list of the elements of $\left \langle {a} \right \rangle$

where $\left \langle {a} \right \rangle$ is the cyclic group generated by $a$.

Order of Subgroup Generated by Single Element

$\left|{\left \langle {a} \right \rangle}\right| = k$

where $\left \langle {a} \right \rangle$ is the smallest subgroup of $G$ containing $a$.

That is, the order of the subgroup generated by $a$ is equal to the order of $a$.

(Some sources use this as the definition of the order of an element, and from it derive Order of an Element.)