# Order of a group element

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 $$1$$.

# Examples

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.

Parents:

• Group

The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.