Order of a group

The or­der \(|G|\) of a group \(G\) is the size of its un­der­ly­ing set. For ex­am­ple, if \(G=(X,\bullet)\) and \(X\) has nine el­e­ments, we say that \(G\) has or­der \(9\). If \(X\) is in­finite, we say \(G\) is in­finite; if \(X\) is finite, we say \(G\) is finite.

The or­der of an el­e­ment \(g \in G\) of a group is the small­est non­nega­tive in­te­ger \(n\) such that \(g^n = e\), or \(\infty\) if there is no such in­te­ger. The re­la­tion­ship be­tween this us­age of or­der and the above us­age of or­der is that the or­der of \(g \in G\) in this sense is the or­der of the sub­group \(\langle g \rangle = \{ 1, g, g^2, \dots \}\) of \(G\) gen­er­ated by \(g\) in the above sense.


  • Group

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