Order (Groups)

order (algebraic)

The order of $a \in G$ for some Group $G$ is the smallest positive integer $n \in N$ such that:

a^{n} = 1

Note that some notations for order are:

{ord}_{G} (a)

You may think having the $G$ subscript isn't necessary, but later we'll be dealing with families of groups so this is helpful. Another option is:

ord (a)

or even:

| a |

where is the absolute value coming from? The idea is that absolute value usually just means distance, and for groups it dictates how many elements in the group. Namely:

| G | = # elements in G

Extended Example: Dihedral Groups

See Dihedral Groups