# 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.