Profinite group (original) (raw)
In der Mathematik ist eine proendliche oder profinite Gruppe eine topologische Gruppe G, die der inverse (projektive) Limes eines Systems von endlichen Gruppen ist. Dieser Limes wird in der Kategorie der topologischen Gruppen gebildet; hierbei betrachtet man jede endliche Gruppe als topologische Gruppe mit der diskreten Topologie. Eine topologische Gruppe ist genau dann proendlich, wenn sie Hausdorffsch, kompakt und total unzusammenhängend ist.
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| dbo:abstract | In der Mathematik ist eine proendliche oder profinite Gruppe eine topologische Gruppe G, die der inverse (projektive) Limes eines Systems von endlichen Gruppen ist. Dieser Limes wird in der Kategorie der topologischen Gruppen gebildet; hierbei betrachtet man jede endliche Gruppe als topologische Gruppe mit der diskreten Topologie. Eine topologische Gruppe ist genau dann proendlich, wenn sie Hausdorffsch, kompakt und total unzusammenhängend ist. (de) En matemática, un grupo pro-finito G es un grupo que, en cierto modo, está muy "próximo" a ser finito. (es) In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists such that every group in the system can be generated by elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profinite group. Important examples of profinite groups are the additive groups of p-adic integers and the Galois groups of infinite-degree field extensions. Every profinite group is compact and totally disconnected. A non-compact generalization of the concept is that of locally profinite groups. Even more general are the totally disconnected groups. (en) En théorie des groupes, un groupe profini est un groupe topologique obtenu comme limite projective de groupes finis discrets. La notion de groupe profini est particulièrement utile en théorie de Galois, pour pouvoir travailler avec des extensions infinies. Comme plus généralement en théorie des catégories, cette limite projective est uniquement définie à unique isomorphisme près. Elle peut être interprétée comme objet final d'une bonne catégorie. (fr) 수학에서 사유한군(射有限群, 영어: profinite group)은 유한군의 으로 얻어지는 위상군이다. (ko) In matematica, un gruppo profinito è un gruppo topologico che si può costruire con un certo processo di limite a partire da gruppi finiti. Molti teoremi validi per i gruppi finiti, quali i teoremi di Sylow, ammettono generalizzazioni naturali ai gruppi profiniti. Formalmente, un gruppo profinito si può definire come un gruppo topologico T2, compatto con un sistema di intorni di fatto di sottogruppi normali. (it) 数学において射有限群（しゃゆうげんぐん、英語: pro-finite group）あるいは副有限群（ふくゆうげんぐん）は、有限群の射影系の極限になっているような位相群である。ガロア群やp-進整数を係数とする代数群など、数論的に興味深い様々な群が射有限群の構造を持つ。 射有限群は完全不連結でコンパクトなハウスドルフ位相群として定義される。同値な定義として、離散有限群の成す射影系（逆系）の射影極限（逆極限）として得られる位相群に同型であるような群を射有限群と定めるいうこともできる。 (ja) In de groepentheorie, een deelgebied van de wiskunde, is een profiniete groep een topologische groep die in zekere zin wordt samengesteld uit eindige groepen. Profinieet groepen hebben veel eigenschappen gemeen met hun eindige quotiënten. (nl) У математиці проскінченною групою називається топологічна група, яка є проєктивною границею скінченних груп. Для них існують узагальнення багатьох властивостей скінченних груп, зокрема теореми Лагранжа і Силова. Некомпактним узагальненням проскінченних груп є локально проскінченні групи. (uk) Проконечная группа — топологическая группа, являющаяся проективным пределом системы конечных групп , , снабженных дискретной топологией ( — предупорядоченное множество). (ru) |
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| rdfs:comment | In der Mathematik ist eine proendliche oder profinite Gruppe eine topologische Gruppe G, die der inverse (projektive) Limes eines Systems von endlichen Gruppen ist. Dieser Limes wird in der Kategorie der topologischen Gruppen gebildet; hierbei betrachtet man jede endliche Gruppe als topologische Gruppe mit der diskreten Topologie. Eine topologische Gruppe ist genau dann proendlich, wenn sie Hausdorffsch, kompakt und total unzusammenhängend ist. (de) En matemática, un grupo pro-finito G es un grupo que, en cierto modo, está muy "próximo" a ser finito. (es) En théorie des groupes, un groupe profini est un groupe topologique obtenu comme limite projective de groupes finis discrets. La notion de groupe profini est particulièrement utile en théorie de Galois, pour pouvoir travailler avec des extensions infinies. Comme plus généralement en théorie des catégories, cette limite projective est uniquement définie à unique isomorphisme près. Elle peut être interprétée comme objet final d'une bonne catégorie. (fr) 수학에서 사유한군(射有限群, 영어: profinite group)은 유한군의 으로 얻어지는 위상군이다. (ko) In matematica, un gruppo profinito è un gruppo topologico che si può costruire con un certo processo di limite a partire da gruppi finiti. Molti teoremi validi per i gruppi finiti, quali i teoremi di Sylow, ammettono generalizzazioni naturali ai gruppi profiniti. Formalmente, un gruppo profinito si può definire come un gruppo topologico T2, compatto con un sistema di intorni di fatto di sottogruppi normali. (it) 数学において射有限群（しゃゆうげんぐん、英語: pro-finite group）あるいは副有限群（ふくゆうげんぐん）は、有限群の射影系の極限になっているような位相群である。ガロア群やp-進整数を係数とする代数群など、数論的に興味深い様々な群が射有限群の構造を持つ。 射有限群は完全不連結でコンパクトなハウスドルフ位相群として定義される。同値な定義として、離散有限群の成す射影系（逆系）の射影極限（逆極限）として得られる位相群に同型であるような群を射有限群と定めるいうこともできる。 (ja) In de groepentheorie, een deelgebied van de wiskunde, is een profiniete groep een topologische groep die in zekere zin wordt samengesteld uit eindige groepen. Profinieet groepen hebben veel eigenschappen gemeen met hun eindige quotiënten. (nl) У математиці проскінченною групою називається топологічна група, яка є проєктивною границею скінченних груп. Для них існують узагальнення багатьох властивостей скінченних груп, зокрема теореми Лагранжа і Силова. Некомпактним узагальненням проскінченних груп є локально проскінченні групи. (uk) Проконечная группа — топологическая группа, являющаяся проективным пределом системы конечных групп , , снабженных дискретной топологией ( — предупорядоченное множество). (ru) In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists such that every group in the system can be generated by elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. (en) |
| rdfs:label | Proendliche Gruppe (de) Grupo profinito (es) Groupe profini (fr) Gruppo profinito (it) 사유한군 (ko) 射有限群 (ja) Profinite group (en) Profiniete groep (nl) Проконечная группа (ru) Проскінченна група (uk) |
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