Results concerning Order of Element
Jump to navigation
Jump to search
Theorems
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 $\gen a$