# 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.