{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]], let $\\pi \\colon G \\to G^{\\mathrm{ab}}$ be the projection to its [[abelianisation|0AQZV0905N]], and let $A$ be an [[abelian group|082GHNPJ53]].\nFor each [[group homomorphism|0JYTG5K9Y4]] $\\varphi \\colon G \\to A$, there is exactly one group homomorphism $\\bar{\\varphi} \\colon G^{\\mathrm{ab}} \\to A$ with $\\bar{\\varphi} \\circ \\pi = \\varphi$.\n\n## Remarks\n\nEach commutator goes to $e$ under $\\varphi$, because $A$ is abelian, so $[G, G] \\subseteq \\ker \\varphi$.\nThen the [[universal property of the quotient group|0GJ83E78A5]] gives $\\bar{\\varphi}$.\nThus the homomorphisms from $G$ to $A$ correspond exactly to the homomorphisms from $G^{\\mathrm{ab}}$ to $A$.\n\n## Explanations\n\n### L1 human\n\nA homomorphism into an abelian group cannot see the commutators of $G$.\nSo it factors through the abelianisation, where the commutators are gone.\nThis makes the abelianisation the best abelian approximation of $G$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "20F12"
    ],
    "name": "Universal property of the abelianisation",
    "role": "theorem",
    "sources": [
      {
        "locator": "Abelianization",
        "url": "https://en.wikipedia.org/wiki/Commutator_subgroup"
      }
    ],
    "tag": "0EJ4PC8NJ6",
    "uses": [
      "0H9K5D3RDV",
      "0GJ83E78A5",
      "0MS8QGA6S5"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0EJ4PC8NJ6/",
  "links": [
    {
      "field": "uses",
      "target": "0H9K5D3RDV"
    },
    {
      "field": "uses",
      "target": "0GJ83E78A5"
    },
    {
      "field": "uses",
      "target": "0MS8QGA6S5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0EJ4PC8NJ6.md",
  "msc": [
    "20F12"
  ],
  "name": "Universal property of the abelianisation",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0EJ/0EJ4PC8NJ6.md",
  "status": null,
  "tag": "0EJ4PC8NJ6",
  "uri": "m2://0EJ4PC8NJ6",
  "url": "https://math2.tamnd.com/wiki/universal-property-of-the-abelianisation/",
  "writing": {
    "tag": "0EJ4PC8NJ6",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}