Group

Definition: Group

A group $G$ is a set equipped with a binary operation $(a,b)\mapsto a\cdot b$ satisfying the following properties:

• Associativity: $a,b,c(a\cdot b)\cdot c=a\cdot (b\cdot c)$
• Identity: $\exists e\in G\text{ s.t. }a\cdot e=a=e\cdot a$
• Inverse: Given $a\in G$, $\exists b\in G\text{ s.t. }a\cdot b=e=b\cdot a$

Order

Definition: Order

The order of a group, denoted $\left|G\right|$, is the cardinality of $G$

Definition: Order (element)

If $a\in G$, then the order of $a$ is the smallest positive integer $n$ such that $a^n=e$. This is equivalent to the order of the Subgroup generated by $a$.