{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a [[cyclic group|0KVGH0J5B3]].\nThen $G$ is [[abelian|082GHNPJ53]].\n\n## Remarks\n\nIf $g$ generates $G$, then any two elements are $g^a$ and $g^b$ for integers $a, b$, and $g^a g^b = g^{a+b} = g^b g^a$ by [[the laws of exponents|00DRK90Z6K]].\nThe converse is false: the Klein four-group is abelian but not cyclic.\nThe group $(\\mathbb{Q}, +)$ is abelian but not cyclic.\n\n## Explanations\n\n### L1 human\n\nAll elements of a cyclic group are powers of one element.\nTwo powers of the same element always commute, because the exponents just add.\nThus the order of the factors never matters in a cyclic group.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "20A05"
    ],
    "name": "Cyclic groups are abelian",
    "role": "proposition",
    "sources": [
      {
        "locator": "Additional properties",
        "url": "https://en.wikipedia.org/wiki/Cyclic_group"
      }
    ],
    "tag": "04RG02S6BA",
    "uses": [
      "0B85Q41RP6",
      "00DRK90Z6K",
      "071DDFVG4J"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/04RG02S6BA/",
  "links": [
    {
      "field": "uses",
      "target": "0B85Q41RP6"
    },
    {
      "field": "uses",
      "target": "00DRK90Z6K"
    },
    {
      "field": "uses",
      "target": "071DDFVG4J"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/04RG02S6BA.md",
  "msc": [
    "20A05"
  ],
  "name": "Cyclic groups are abelian",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/04R/04RG02S6BA.md",
  "status": null,
  "tag": "04RG02S6BA",
  "uri": "m2://04RG02S6BA",
  "url": "https://math2.tamnd.com/wiki/cyclic-groups-are-abelian/",
  "writing": {
    "tag": "04RG02S6BA",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}