{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "D8A4VEQR98"
    },
    {
      "field": "instance_of",
      "target": "F0XAMM2PPS"
    },
    {
      "field": "defines",
      "target": "S8Y8RVZY79"
    }
  ],
  "body": "\n## Summary\n\nA divisible group is an [[abelian group|082GHNPJ53]] in which every element can be divided by every positive integer.\nThe rational numbers, the real numbers and $\\Q/\\Z$ are divisible; the integers are not.\nDivisible groups are exactly the [[injective|85H7PRHDPZ]] $\\Z$-modules, and every divisible group is a direct sum of copies of $\\Q$ and of [[Prüfer groups|F0XAMM2PPS]].\nEvery abelian group embeds in a divisible group, and a divisible subgroup is always a direct summand.\n\n## Explanations\n\n### L1 human\n\nIn $\\Q$, every number can be halved, divided by three, and so on, and the answer stays in $\\Q$.\nA divisible group is an abelian group with this property for every element and every positive integer.\nIn $\\Z$ this fails, since $1$ has no half.\nOn the circle of angles, every angle can be divided by $n$, so the circle group is divisible too.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "20K40",
      "20K25"
    ],
    "name": "Divisible group",
    "same_as": {
      "wikidata": "Q1782332",
      "wikipedia": "Divisible group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Divisible_group"
      }
    ],
    "specializes": [
      "082GHNPJ53"
    ],
    "tag": "ZXVBEPMTCN"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ZXVBEPMTCN/",
  "links": [
    {
      "field": "specializes",
      "target": "082GHNPJ53"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZXVBEPMTCN.md",
  "msc": [
    "20K40",
    "20K25"
  ],
  "name": "Divisible group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZXV/ZXVBEPMTCN.md",
  "status": null,
  "tag": "ZXVBEPMTCN",
  "uri": "m2://ZXVBEPMTCN",
  "url": "https://math2.tamnd.com/wiki/divisible-group/",
  "writing": {
    "tag": "ZXVBEPMTCN",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}