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  "body": "\n## Definition\n\nA [[group|G5S1M9]] $G$ is cyclic when there is an element $g \\in G$ with $G = \\langle g \\rangle$, where $\\langle g \\rangle$ is the [[cyclic subgroup generated by $g$|08ABT85339]].\nSuch an element $g$ is called a generator of $G$.\n\n## Remarks\n\nThe group $(\\mathbb{Z}, +)$ is cyclic, and its generators are $1$ and $-1$.\nThe integers modulo $n$ under addition form a cyclic group of order $n$, generated by $1$.\nA cyclic group can have many generators: in the integers modulo $n$, the generators are the residues coprime to $n$.\nThe [[circle group|04EAFVFTF7]] is not cyclic, because it is uncountable.\n\n## Explanations\n\n### L1 human\n\nA cyclic group is a group in which each element is a power of one fixed element.\nThe rotations of a regular polygon with $n$ sides form a cyclic group: each rotation is a power of the smallest rotation.\nThe word \"cyclic\" comes from this picture of going around a circle.\n",
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