{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0E088ZP533"
    },
    {
      "field": "uses",
      "target": "0RFP7F3SEW"
    }
  ],
  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nA composition series of $G$ is a [[subnormal series|0ZZR8D7Y1Z]] $\\{e\\} = G_0 \\trianglelefteq G_1 \\trianglelefteq \\cdots \\trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is a [[simple group|0YMYMNQ0QZ]].\n\n## Remarks\n\nEquivalently, the inclusions are strict and each $G_{i-1}$ is a maximal proper normal subgroup of $G_i$.\nEach finite group has a composition series, but the group $\\mathbb{Z}$ has none.\nFor example, $\\{0\\} \\trianglelefteq \\langle 6 \\rangle \\trianglelefteq \\langle 3 \\rangle \\trianglelefteq \\mathbb{Z}/12\\mathbb{Z}$ and $\\{0\\} \\trianglelefteq \\langle 6 \\rangle \\trianglelefteq \\langle 2 \\rangle \\trianglelefteq \\mathbb{Z}/12\\mathbb{Z}$ are composition series.\n\n## Explanations\n\n### L1 human\n\nA composition series cuts a group into layers that cannot be cut further.\nEach layer is a simple group.\nIt is like the factorisation of an integer into prime numbers.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0XV2A49X6Q",
    "kind": "definition",
    "msc": [
      "20E15",
      "20D30"
    ],
    "name": "Composition series",
    "sources": [
      {
        "locator": "For groups",
        "url": "https://en.wikipedia.org/wiki/Composition_series"
      }
    ],
    "tag": "0NB8VPB93V",
    "uses": [
      "096HN851YZ",
      "0C093D1T4M"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0NB8VPB93V/",
  "links": [
    {
      "field": "defines",
      "target": "0XV2A49X6Q"
    },
    {
      "field": "uses",
      "target": "096HN851YZ"
    },
    {
      "field": "uses",
      "target": "0C093D1T4M"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0NB8VPB93V.md",
  "msc": [
    "20E15",
    "20D30"
  ],
  "name": "Composition series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0NB/0NB8VPB93V.md",
  "status": null,
  "tag": "0NB8VPB93V",
  "uri": "m2://0NB8VPB93V",
  "url": "https://math2.tamnd.com/wiki/composition-series-definition/",
  "writing": {
    "tag": "0NB8VPB93V",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}