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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

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rdf:type	Thing
rdfs:label	قائمة مواضيع نظرية الزمر (ar) Daftar topik teori grup (in) Glossario di teoria dei gruppi (it) Lexique des groupes (fr) List of group theory topics (en) 群論の用語 (ja) Глоссарий теории групп (ru) Словник термінів теорії груп (uk)
rdfs:comment	Un gruppo è un insieme munito di un'operazione associativa dotata di elemento neutro e tale che ogni elemento possiede un inverso. Gruppi molto importanti sono costituiti da trasformazioni; altri gruppi che si incontrano spesso sono costituiti da insiemi numerici muniti della moltiplicazione. In genere l'operazione di un gruppo viene chiamata prodotto e il suo elemento neutro viene detto unità o elemento identità. In questo articolo useremo e per denotare l'unità di un gruppo. (it) 群 (G, •) は集合 G で三つの公理を満たす G 上の（つまり G において閉じた）二項演算 "•" を組にしたものである。群の三公理とは * 演算の結合律: G の任意の元 a, b, c に対して (a • b) • c = a • (b • c) が成り立つ。 * 単位元の存在: e ∈ G が存在して、G のいかなる元 a に対しても e • a = a • e = a を満たす。 * 逆元の存在: G のそれぞれの元 a に対して a • b = b • a = e を満たす G の元 b が存在する。ここで e は単位元。群の簡単な例としては整数全体の集合 Z に加法を考えたもの、0 でない有理数全体の成す集合 Q× に乗法を考えたものが挙げられる。後者はもっと一般に任意の環 R について、その単元の全体 R× は乗法群を成す。群の定義および更なる例については群の項に譲る。しかし群には上に挙げたものよりももっと一般の構造が含まれる。群論とは、問題とする群の演算や元が具体的にどのような姿をしているかということに依存しない、群についての抽象的な言明を与えることを目的とするものである。この用語集では群論において広く用いられる基本的な概念についての短い説明を提供する。群論の話題についての一般の記述は群論の項を参照されたい。また、群論の話題一覧等も参照のこと。 (ja) Для загального ознайомлення з теорією груп див. Група (математика) і Теорія груп. Курсив позначає посилання на цей словник. (uk) В этой статье приведены основные термины, используемые в теории групп. Курсив обозначает внутреннюю ссылку на данный глоссарий. В конце приводится таблица основных обозначений, применяемых в теории групп. (ru) In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. (en)
rdfs:seeAlso	List of representation theory topics
dcterms:subject	Group theory Properties of groups Mathematics-related lists Outlines of mathematics and logic Wikipedia outlines
Wikipage page ID	100303 (xsd:integer)
Wikipage revision ID	1088030308 (xsd:integer)
Link from a Wikipage to another Wikipage	Camille Jordan Baby Monster group Presentation of a group Projective representation Public key cryptography Quandle Quaternion Roger Carter (mathematician) List of abstract algebra topics List of finite simple groups Module (mathematics) Monster group Monoid ring Nielsen transformation Representation theory Butterfly lemma Bernd Fischer (mathematician) Bernhard Neumann Bilinear operator Braid group David Hilbert Derived group Algebraic group Algebraic structure Algebraic topology Homogeneous space Homomorphism Hydrogen atom Hyperbolic group Joseph-Louis Lagrange Bertram Huppert Pauli matrices Permutation Peter Ludwig Mejdell Sylow Reinhold Baer Richard Brauer Richard Dedekind Charles Leedham-Green Burnside's problem Vector space Jakob Nielsen (mathematician) Number Lie group Isomorphism theorem Multiplicative inverse Robert Steinberg Group theory Commutative Commutator Complete group Conjugate closure Continuous symmetry Coset Cryptography Maschke's theorem Mathematics Matrix (mathematics) Max August Zorn Gell-Mann matrices General linear group Geometric group theory Non-abelian group Normal subgroup Simple group Quasigroup Quotient group Tarski monster group Elliptic curve Emmy Noether Galois theory Generalized Fitting subgroup Generating set of a group Geometry George Abram Miller George Glauberman Giovanni Frattini Graham Higman Minkowski's theorem

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