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text: '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 $e$.\n\n%%%knows-requisite([3gg]):\nEquivalently, 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 $+$.\n%%%\n\nConventionally, the identity element itself has order $1$.\n\n# Examples\n\n%%%knows-requisite([497]):\nIn the [-497] $S_5$, the order of an element is the [-least_common_multiple] of its [4cg cycle type].\n%%%\n%%%knows-requisite([47y]):\nIn the [-47y] $C_6$, the order of the generator is $6$.\nIf we view $C_6$ as being the integers [modular_arithmetic 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 $2$.\n%%%\n\nIn the group $\\mathbb{Z}$ of [48l integers] under addition, every element except $0$ has infinite order. $0$ itself has order $1$, being the identity.',
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