{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "64K1A69C80"
    },
    {
      "field": "uses",
      "target": "64K1A69C80"
    },
    {
      "field": "defines",
      "target": "JZ7CSKTZ16"
    }
  ],
  "body": "\n## Summary\n\nA group extension is a group $G$ with a [[normal subgroup|0CB0DX5BVV]] $N$ and a [[quotient group|0KYX2N69J0]] $G/N$ isomorphic to a given group $Q$.\nIt is the way to build a group out of two smaller pieces.\nThe extension problem asks for all groups $G$ that can be built from given $N$ and $Q$.\nEvery finite group is built from simple groups by repeated extensions, along a [[composition series|0XV2A49X6Q]].\n\n## Explanations\n\n### L1 human\n\nA group extension glues a group $N$ and a group $Q$ together into a bigger group $G$.\nInside $G$ you find a copy of $N$ as a normal subgroup, and when you collapse that copy to a point, what is left is $Q$.\nThe same two pieces can be glued in different ways.\nFor example, both $\\Z/4\\Z$ and $\\Z/2\\Z \\times \\Z/2\\Z$ have a subgroup of order $2$ with quotient of order $2$.\nSo both are extensions of a group of order $2$ by a group of order $2$, but they are not isomorphic.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "20E22",
      "20J06"
    ],
    "name": "Group extension",
    "same_as": {
      "wikidata": "Q3041173",
      "wikipedia": "Group extension"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Group_extension"
      }
    ],
    "tag": "MF4N3VGGFV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/MF4N3VGGFV/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/MF4N3VGGFV.md",
  "msc": [
    "20E22",
    "20J06"
  ],
  "name": "Group extension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MF4/MF4N3VGGFV.md",
  "status": null,
  "tag": "MF4N3VGGFV",
  "uri": "m2://MF4N3VGGFV",
  "url": "https://math2.tamnd.com/wiki/group-extension/",
  "writing": {
    "tag": "MF4N3VGGFV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}