Given an element 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 $e$ .

Equivalently, it is the order of the group $⟨ g ⟩$ 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

In the symmetric group $S_{5}$ , the order of an element is the least_common_multiple of its cycle type.

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 $Z$ of integers under addition, every element except $0$ has infinite order. $0$ itself has order $1$ , being the identity.

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Order of a group element

Examples