Order of a group element


by Patrick Stevens Jun 15 2016

Given an element $~$g$~$ of group $~$(G, +)$~$ (which henceforth we abbreviate simply as $~$G$~$), the order of $~$g$~$ is the number of times we must add $~$g$~$ to itself to obtain the identity element $~$e$~$.

%%%knows-requisite(Order of a group): Equivalently, it is the order of the group $~$\langle g \rangle$~$ generated by $~$g$~$: that is, the order of $~$\{ e, g, g^2, \dots, g^{-1}, g^{-2}, \dots \}$~$ under the inherited group operation $~$+$~$. %%%

Conventionally, the identity element itself has order $~$1$~$.


%%%knows-requisite(Symmetric group): In the Symmetric group $~$S_5$~$, the order of an element is the Least common multiple of its cycle type. %%% %%%knows-requisite(Cyclic group): In the Cyclic group $~$C_6$~$, the order of the generator is $~$6$~$. If we view $~$C_6$~$ as being the integers modulo $~$6$~$ under addition, then the element $~$0$~$ has order $~$1$~$; the elements $~$1$~$ and $~$5$~$ have order $~$6$~$; the elements $~$2$~$ and $~$4$~$ have order $~$3$~$; and the element $~$3$~$ has order $~$2$~$. %%%

In the group $~$\mathbb{Z}$~$ of integers under addition, every element except $~$0$~$ has infinite order. $~$0$~$ itself has order $~$1$~$, being the identity.