Definition (Group)

A group is a pair $(G,\cdot)$ where $G$ is a set and $\cdot$ is a binary operation on elements of $G$ such that:

1. Closure: $$g,h\in G\implies g\cdot h\in G$$
2. Associativity: $$(g\cdot h)\cdot k=g\cdot(h\cdot k)$$
3. Existence of Identity: $$\exists1:1\cdot g=g$$
4. Existence of Inverse: $\forall g\in G,\exists g^{-1}\in G$ such that $$g^{-1}\cdot g=1$$

Definition (Order)

Given a group $G$, the order is the smallest integer $r$ for any element $x\in G$ s.t. $$x^{r}=1$$