{
  "id": "enwiki-001378786148",
  "slug": "subnormal-subgroup",
  "title": "Subnormal subgroup",
  "qid": "",
  "lead": "In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G. In notation, H is k -subnormal in G if there are subgroups : H=H_0,H_1,H_2,\\ldots, H_k=G of G such that H_i is normal in H_{i+1} for each i . A subnormal subgroup is a subgroup that is k -subnormal for some positive integer k . Some facts about subnormal subgroups: * A 1-subnormal subgroup is a proper normal subgroup (and vice versa). * A finitely generated group is nilpotent if and only if each of its subgroups is subnormal. * Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal. * Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal. * Every 2-subnormal subgroup is a conjugate-permutable subgroup. * A subgroup H is subnormal in G if and only if |HK| divides |G| for every subgroup K of G . The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality. If every subnormal subgroup of G is normal in G, then G is called a T-group.",
  "body": "In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G. In notation, H is k -subnormal in G if there are subgroups H=H_0,H_1,H_2,\\ldots, H_k=G of G such that H_i is normal in H_{i+1} for each i . A subnormal subgroup is a subgroup that is k -subnormal for some positive integer k . Some facts about subnormal subgroups: A 1-subnormal subgroup is a proper normal subgroup (and vice versa). A finitely generated group is nilpotent if and only if each of its subgroups is subnormal. Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal. Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal. Every 2-subnormal subgroup is a conjugate-permutable subgroup. A subgroup H is subnormal in G if and only if |HK| divides |G| for every subgroup K of G . The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality. If every subnormal subgroup of G is normal in G, then G is called a T-group. ==See also== Characteristic subgroup Normal core Normal closure Ascendant subgroup Descendant subgroup Serial subgroup ==References==",
  "categories": [
    "29176650"
  ],
  "license": "CC BY-SA 4.0",
  "source_uri": "https://en.wikipedia.org/w/index.php?oldid=1378786148",
  "content_hash": "",
  "created_at": null,
  "updated_at": 1791251434.0,
  "claims": [],
  "contradictions": [],
  "confidence": 0.6,
  "csl": {
    "id": "enwiki-001378786148",
    "type": "webpage",
    "title": "Subnormal subgroup",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/subnormal-subgroup",
    "issued": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "accessed": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001378786148",
    "isBasedOn": "https://en.wikipedia.org/w/index.php?oldid=1378786148",
    "note": "Immutable snapshot enwiki-001378786148"
  }
}