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  "body": "\n## Example\n\nThe [[additive group of integers|0BSCJA5G53]] $(\\mathbb{Z}, +)$ is an infinite [[cyclic group|0KVGH0J5B3]].\nThe element $1$ generates it, because each integer is a sum of copies of $1$ or of $-1$.\nIts only generators are $1$ and $-1$.\n\n## Remarks\n\nIf $G = \\langle g \\rangle$ is an infinite cyclic group, then the powers $g^k$ for $k \\in \\mathbb{Z}$ are all distinct.\nThen $k \\mapsto g^k$ is an isomorphism from $\\mathbb{Z}$ to $G$, thus each infinite cyclic group is isomorphic to $(\\mathbb{Z}, +)$.\nThe group of the translations of the plane by integer multiples of a fixed vector is an infinite cyclic group.\n\n## Explanations\n\n### L1 human\n\nIn an infinite cyclic group, the powers of the generator never repeat.\nYou can walk forever in both directions without coming back.\nThis is exactly how the integers behave under addition.\n",
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