See abelian group in All languages combined, or Wiktionary
{
"etymology_text": "Named in honour of Niels Henrik Abel (1802–1829), a Norwegian mathematician.",
"forms": [
{
"form": "abelian groups",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "abelian group (plural abelian groups)",
"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": "other",
"name": "Entries with translation boxes",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Japanese terms with redundant script codes",
"parents": [
"Terms with redundant script codes",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Mandarin terms with redundant transliterations",
"parents": [
"Terms with redundant transliterations",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Czech translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Dutch translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Esperanto translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Estonian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Finnish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with French translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with German translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Greek translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Hungarian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Icelandic translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Italian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Japanese translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Mandarin translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Norwegian Bokmål translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Polish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Portuguese translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Romanian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Russian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Serbo-Croatian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Spanish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Swedish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Telugu translations",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"derived": [
{
"word": "free abelian group"
}
],
"examples": [
{
"text": "1986, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, 2010 softcover reprint, page 12,\nLet G and H be partially ordered abelian groups. A positive homomorphism from G to H is any abelian group homomorphism f:G→H that maps positive elements to positive elements, that is, f(G⁺)⊆H⁺."
},
{
"text": "2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, Softcover reprint of 1st edition, page 74,\nChapter 2 is a brief introduction to some fundamental techniques for countable torsion-free abelian groups."
},
{
"ref": "2013, Karl H. Hofmann, Sidney A. Morris, The Structure of Compact Groups: A Primer for the Student: A Handbook for the Expert, Walter de Gruyter, page 299:",
"text": "By the end of Chapter 2 we had the full power of the Pontryagin Duality Theorem for compact abelian groups and for discrete abelian groups. Locally compact abelian groups are much closer to compact abelian groups than is apparent at first sight.",
"type": "quote"
}
],
"glosses": [
"A group in which the group operation is commutative."
],
"hypernyms": [
{
"word": "group"
},
{
"word": "solvable group"
}
],
"hyponyms": [
{
"word": "cyclic group"
},
{
"word": "free abelian group"
},
{
"word": "torsion subgroup"
}
],
"id": "en-abelian_group-en-noun-KMNnpulx",
"links": [
[
"algebra",
"algebra"
],
[
"group",
"group"
],
[
"commutative",
"commutative"
]
],
"raw_glosses": [
"(algebra) A group in which the group operation is commutative."
],
"synonyms": [
{
"word": "commutative group"
},
{
"word": "Abelian group"
}
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "group in which the group operation is commutative",
"word": "阿貝爾群"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "Ābèi'ěr qún",
"sense": "group in which the group operation is commutative",
"word": "阿贝尔群"
},
{
"code": "cs",
"lang": "Czech",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "abelovská grupa"
},
{
"code": "nl",
"lang": "Dutch",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "Abelse groep"
},
{
"code": "nl",
"lang": "Dutch",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "Abelse ring"
},
{
"code": "eo",
"lang": "Esperanto",
"sense": "group in which the group operation is commutative",
"word": "abela grupo"
},
{
"code": "eo",
"lang": "Esperanto",
"sense": "group in which the group operation is commutative",
"word": "komuta grupo"
},
{
"code": "et",
"lang": "Estonian",
"sense": "group in which the group operation is commutative",
"word": "Abeli rühm"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "group in which the group operation is commutative",
"word": "Abelin ryhmä"
},
{
"code": "fr",
"lang": "French",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "groupe abélien"
},
{
"code": "fr",
"lang": "French",
"sense": "group in which the group operation is commutative",
"word": "groupe commutatif"
},
{
"code": "de",
"lang": "German",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "abelsche Gruppe"
},
{
"code": "de",
"lang": "German",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "kommutative Gruppe"
},
{
"code": "el",
"lang": "Greek",
"roman": "avelianí omáda",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "αβελιανή ομάδα"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "group in which the group operation is commutative",
"word": "Abel-csoport"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "Abel-grúpa"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "víxlgrúpa"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "abelsk grúpa"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "víxlin grúpa"
},
{
"code": "it",
"lang": "Italian",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "gruppo abeliano"
},
{
"code": "ja",
"lang": "Japanese",
"roman": "Āberu-gun",
"sense": "group in which the group operation is commutative",
"word": "アーベル群"
},
{
"code": "nb",
"lang": "Norwegian Bokmål",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine",
"masculine"
],
"word": "abelsk gruppe"
},
{
"code": "pl",
"lang": "Polish",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "grupa przemienna"
},
{
"code": "pl",
"lang": "Polish",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "grupa abelowa"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "grupo abeliano"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "group in which the group operation is commutative",
"tags": [
"neuter"
],
"word": "grup abelian"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "group in which the group operation is commutative",
"tags": [
"neuter"
],
"word": "grup comutativ"
},
{
"code": "ru",
"lang": "Russian",
"roman": "ábeleva grúppa",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "а́белева гру́ппа"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "Abelova grupa"
},
{
"code": "es",
"lang": "Spanish",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "grupo abeliano"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "group in which the group operation is commutative",
"tags": [
"common-gender"
],
"word": "abelsk grupp"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "group in which the group operation is commutative",
"tags": [
"common-gender"
],
"word": "kommutativ grupp"
},
{
"code": "te",
"lang": "Telugu",
"roman": "ebiliyan samūhaṁ",
"sense": "group in which the group operation is commutative",
"word": "ఎబిలియన్ సమూహం"
}
],
"wikipedia": [
"Niels Henrik Abel",
"abelian group"
]
}
],
"sounds": [
{
"ipa": "/əˈbi.li.ən ɡɹup/",
"tags": [
"US"
]
},
{
"ipa": "/əˈbil.jən ɡɹup/",
"tags": [
"US"
]
}
],
"word": "abelian group"
}
{
"derived": [
{
"word": "free abelian group"
}
],
"etymology_text": "Named in honour of Niels Henrik Abel (1802–1829), a Norwegian mathematician.",
"forms": [
{
"form": "abelian groups",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "abelian group (plural abelian groups)",
"name": "en-noun"
}
],
"hypernyms": [
{
"word": "group"
},
{
"word": "solvable group"
}
],
"hyponyms": [
{
"word": "cyclic group"
},
{
"word": "free abelian group"
},
{
"word": "torsion subgroup"
}
],
"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",
"Entries with translation boxes",
"Japanese terms with redundant script codes",
"Mandarin terms with redundant transliterations",
"Pages with 1 entry",
"Pages with entries",
"Requests for review of Mandarin translations",
"Terms with Czech translations",
"Terms with Dutch translations",
"Terms with Esperanto translations",
"Terms with Estonian translations",
"Terms with Finnish translations",
"Terms with French translations",
"Terms with German translations",
"Terms with Greek translations",
"Terms with Hungarian translations",
"Terms with Icelandic translations",
"Terms with Italian translations",
"Terms with Japanese translations",
"Terms with Mandarin translations",
"Terms with Norwegian Bokmål translations",
"Terms with Polish translations",
"Terms with Portuguese translations",
"Terms with Romanian translations",
"Terms with Russian translations",
"Terms with Serbo-Croatian translations",
"Terms with Spanish translations",
"Terms with Swedish translations",
"Terms with Telugu translations",
"en:Algebra"
],
"examples": [
{
"text": "1986, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, 2010 softcover reprint, page 12,\nLet G and H be partially ordered abelian groups. A positive homomorphism from G to H is any abelian group homomorphism f:G→H that maps positive elements to positive elements, that is, f(G⁺)⊆H⁺."
},
{
"text": "2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, Softcover reprint of 1st edition, page 74,\nChapter 2 is a brief introduction to some fundamental techniques for countable torsion-free abelian groups."
},
{
"ref": "2013, Karl H. Hofmann, Sidney A. Morris, The Structure of Compact Groups: A Primer for the Student: A Handbook for the Expert, Walter de Gruyter, page 299:",
"text": "By the end of Chapter 2 we had the full power of the Pontryagin Duality Theorem for compact abelian groups and for discrete abelian groups. Locally compact abelian groups are much closer to compact abelian groups than is apparent at first sight.",
"type": "quote"
}
],
"glosses": [
"A group in which the group operation is commutative."
],
"links": [
[
"algebra",
"algebra"
],
[
"group",
"group"
],
[
"commutative",
"commutative"
]
],
"raw_glosses": [
"(algebra) A group in which the group operation is commutative."
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"wikipedia": [
"Niels Henrik Abel",
"abelian group"
]
}
],
"sounds": [
{
"ipa": "/əˈbi.li.ən ɡɹup/",
"tags": [
"US"
]
},
{
"ipa": "/əˈbil.jən ɡɹup/",
"tags": [
"US"
]
}
],
"synonyms": [
{
"word": "commutative group"
},
{
"word": "Abelian group"
}
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "group in which the group operation is commutative",
"word": "阿貝爾群"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "Ābèi'ěr qún",
"sense": "group in which the group operation is commutative",
"word": "阿贝尔群"
},
{
"code": "cs",
"lang": "Czech",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "abelovská grupa"
},
{
"code": "nl",
"lang": "Dutch",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "Abelse groep"
},
{
"code": "nl",
"lang": "Dutch",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "Abelse ring"
},
{
"code": "eo",
"lang": "Esperanto",
"sense": "group in which the group operation is commutative",
"word": "abela grupo"
},
{
"code": "eo",
"lang": "Esperanto",
"sense": "group in which the group operation is commutative",
"word": "komuta grupo"
},
{
"code": "et",
"lang": "Estonian",
"sense": "group in which the group operation is commutative",
"word": "Abeli rühm"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "group in which the group operation is commutative",
"word": "Abelin ryhmä"
},
{
"code": "fr",
"lang": "French",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "groupe abélien"
},
{
"code": "fr",
"lang": "French",
"sense": "group in which the group operation is commutative",
"word": "groupe commutatif"
},
{
"code": "de",
"lang": "German",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "abelsche Gruppe"
},
{
"code": "de",
"lang": "German",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "kommutative Gruppe"
},
{
"code": "el",
"lang": "Greek",
"roman": "avelianí omáda",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "αβελιανή ομάδα"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "group in which the group operation is commutative",
"word": "Abel-csoport"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "Abel-grúpa"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "víxlgrúpa"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "abelsk grúpa"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "víxlin grúpa"
},
{
"code": "it",
"lang": "Italian",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "gruppo abeliano"
},
{
"code": "ja",
"lang": "Japanese",
"roman": "Āberu-gun",
"sense": "group in which the group operation is commutative",
"word": "アーベル群"
},
{
"code": "nb",
"lang": "Norwegian Bokmål",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine",
"masculine"
],
"word": "abelsk gruppe"
},
{
"code": "pl",
"lang": "Polish",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "grupa przemienna"
},
{
"code": "pl",
"lang": "Polish",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "grupa abelowa"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "grupo abeliano"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "group in which the group operation is commutative",
"tags": [
"neuter"
],
"word": "grup abelian"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "group in which the group operation is commutative",
"tags": [
"neuter"
],
"word": "grup comutativ"
},
{
"code": "ru",
"lang": "Russian",
"roman": "ábeleva grúppa",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "а́белева гру́ппа"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "group in which the group operation is commutative",
"tags": [
"feminine"
],
"word": "Abelova grupa"
},
{
"code": "es",
"lang": "Spanish",
"sense": "group in which the group operation is commutative",
"tags": [
"masculine"
],
"word": "grupo abeliano"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "group in which the group operation is commutative",
"tags": [
"common-gender"
],
"word": "abelsk grupp"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "group in which the group operation is commutative",
"tags": [
"common-gender"
],
"word": "kommutativ grupp"
},
{
"code": "te",
"lang": "Telugu",
"roman": "ebiliyan samūhaṁ",
"sense": "group in which the group operation is commutative",
"word": "ఎబిలియన్ సమూహం"
}
],
"word": "abelian group"
}
Download raw JSONL data for abelian group meaning in English (8.0kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.