"semidirect product" meaning in English

{
  "etymology_templates": [
    {
      "args": {
        "1": "en",
        "2": "direct product"
      },
      "expansion": "direct product",
      "name": "m"
    },
    {
      "args": {
        "1": "en",
        "2": "semidirect"
      },
      "expansion": "semidirect",
      "name": "m"
    }
  ],
  "etymology_text": "Refers to the fact that the criteria are less strict than for the direct product; compare semidirect.",
  "forms": [
    {
      "form": "semidirect products",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "semidirect product (plural semidirect products)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
          "parents": [
            "Algebra",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "inner semidirect product"
        },
        {
          "word": "outer semidirect product"
        }
      ],
      "examples": [
        {
          "text": "1998, Hernán Cendra, Darryl D. Holm, Jerrold E. Marsden, Tudor S. Ratiu, Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products, A. G. Khovanskiĭ, A. Varchenko, V. Vassiliev (editors), Geometry of Differential Equations, American Mathematical Society, Translations, Series 2, Volume 186, Advances in the Mathematical Sciences 39, page 8,\nThe preceding result is a special case of a general theorem on reduction by stages for semidirect products acting on a symplectic manifold […] ."
        },
        {
          "ref": "2008, I. Martin Isaacs, Finite Group Theory, American Mathematical Society, page 69",
          "text": "Also, by Lemma 3.1, every group G with a normal subgroup N and complement H is isomorphic to a semidirect product of N by H, and so once we prove Theorem 3.2, it will be fair to say that we have constructed all possible split extensions (up to isomorphism).",
          "type": "quotation"
        },
        {
          "ref": "2012, Fernando Q. Gouvêa, A Guide to Groups, Rings, and Fields, page 61",
          "text": "Theorem 4.8.5 Let G be a group, H,K#x3C;G,H◅ G. Then G is the internal semidirect product of H and K if and only if H#x5C;capK#x3D;1 and G#x3D;HK.\nMany groups can be understood as semidirect products.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup."
      ],
      "id": "en-semidirect_product-en-noun-CqZRJUGG",
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "generalisation",
          "generalisation"
        ],
        [
          "direct product",
          "direct product"
        ],
        [
          "subgroup",
          "subgroup"
        ],
        [
          "normal subgroup",
          "normal subgroup"
        ]
      ],
      "raw_glosses": [
        "(group theory) A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup."
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "semidirect product"
      ]
    }
  ],
  "word": "semidirect product"
}

{
  "derived": [
    {
      "word": "inner semidirect product"
    },
    {
      "word": "outer semidirect product"
    }
  ],
  "etymology_templates": [
    {
      "args": {
        "1": "en",
        "2": "direct product"
      },
      "expansion": "direct product",
      "name": "m"
    },
    {
      "args": {
        "1": "en",
        "2": "semidirect"
      },
      "expansion": "semidirect",
      "name": "m"
    }
  ],
  "etymology_text": "Refers to the fact that the criteria are less strict than for the direct product; compare semidirect.",
  "forms": [
    {
      "form": "semidirect products",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "semidirect product (plural semidirect products)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Group theory"
      ],
      "examples": [
        {
          "text": "1998, Hernán Cendra, Darryl D. Holm, Jerrold E. Marsden, Tudor S. Ratiu, Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products, A. G. Khovanskiĭ, A. Varchenko, V. Vassiliev (editors), Geometry of Differential Equations, American Mathematical Society, Translations, Series 2, Volume 186, Advances in the Mathematical Sciences 39, page 8,\nThe preceding result is a special case of a general theorem on reduction by stages for semidirect products acting on a symplectic manifold […] ."
        },
        {
          "ref": "2008, I. Martin Isaacs, Finite Group Theory, American Mathematical Society, page 69",
          "text": "Also, by Lemma 3.1, every group G with a normal subgroup N and complement H is isomorphic to a semidirect product of N by H, and so once we prove Theorem 3.2, it will be fair to say that we have constructed all possible split extensions (up to isomorphism).",
          "type": "quotation"
        },
        {
          "ref": "2012, Fernando Q. Gouvêa, A Guide to Groups, Rings, and Fields, page 61",
          "text": "Theorem 4.8.5 Let G be a group, H,K#x3C;G,H◅ G. Then G is the internal semidirect product of H and K if and only if H#x5C;capK#x3D;1 and G#x3D;HK.\nMany groups can be understood as semidirect products.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup."
      ],
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "generalisation",
          "generalisation"
        ],
        [
          "direct product",
          "direct product"
        ],
        [
          "subgroup",
          "subgroup"
        ],
        [
          "normal subgroup",
          "normal subgroup"
        ]
      ],
      "raw_glosses": [
        "(group theory) A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup."
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "semidirect product"
      ]
    }
  ],
  "word": "semidirect product"
}