# 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.

Parents:

• Group

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