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.