{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be an [[abelian group|082GHNPJ53]], written additively.\nThe group $A$ is *divisible* if for every $a \\in A$ and every integer $n \\ge 1$ there is $x \\in A$ with $nx = a$.\n\n## Remarks\n\nExample: $\\Q$ is divisible: given $a \\in \\Q$ and $n \\ge 1$, take $x = a/n$.\nExample: $\\Q/\\Z$ is divisible, as a quotient of the divisible group $\\Q$.\n\nNon-example: $\\Z$ is not divisible, since there is no integer $x$ with $2x = 1$.\nA nonzero finite abelian group is not divisible either: if $\\abs{A} = m$ and $a \\neq 0$, then $mx = 0 \\neq a$ for every $x$.\n\nAn abelian group is divisible if and only if it is an [[injective|85H7PRHDPZ]] $\\Z$-module.\nIt is enough to check the condition for $n$ prime.\nEvery divisible group is isomorphic to a direct sum of copies of $\\Q$ and of the [[Prüfer groups|F0XAMM2PPS]] $\\Z(p^\\infty)$, for various primes $p$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "ZXVBEPMTCN",
    "kind": "definition",
    "msc": [
      "20K40",
      "20K25"
    ],
    "name": "Divisible group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Divisible_group"
      }
    ],
    "tag": "S8Y8RVZY79",
    "uses": [
      "082GHNPJ53"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/S8Y8RVZY79/",
  "links": [
    {
      "field": "defines",
      "target": "ZXVBEPMTCN"
    },
    {
      "field": "uses",
      "target": "082GHNPJ53"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/S8Y8RVZY79.md",
  "msc": [
    "20K40",
    "20K25"
  ],
  "name": "Divisible group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S8Y/S8Y8RVZY79.md",
  "status": null,
  "tag": "S8Y8RVZY79",
  "uri": "m2://S8Y8RVZY79",
  "url": "https://math2.tamnd.com/wiki/divisible-group-definition/",
  "writing": {
    "tag": "S8Y8RVZY79",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}