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• Mathematics
  ...etic''' is a branch of mathematics and their basic properties under the [[operation]]s of [[addition]], [[subtraction]], [[multiplication]] and [[division]] an ...tions of the normal operations seen arithmetic and high school algebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common ob
  6 KB (902 words) - 12:53, 3 September 2019
• Commutative property
  An [[operation]] (especially a [[binary operation]]) is said to have the '''commutative property''' or to ''be commutative'' ...]]s, etc.) because <math>\displaystyle a + b = b + a</math>. However, the operation of [[division]] is not commutative over these sets because usually <math>\f
  2 KB (301 words) - 17:46, 16 March 2012
• Permutation
  ==The Symmetric Group== ...>S_n</math> forms a [[group]], known as the [[Symmetric group]], under the operation of permutation composition.
  3 KB (422 words) - 11:01, 25 December 2020
• Negative number
  ...to include the [[complex number]]s (and, more generally, to any additive [[group]]). The negative of a negative real number is a [[positive number]]. The * [[Inverse with respect to an operation]]
  635 bytes (93 words) - 14:07, 16 January 2023
• Ring
  ...ield]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multiplication and addition and denoted <math>\cdot</mat * <math>(R,+)</math> is an [[abelian group]];
  6 KB (994 words) - 06:16, 8 April 2015
• Field
  A '''field''' is a structure in [[abstract algebra]], similar to a [[group]] or a [[ring]]. Informally, fields are the general structure in which the ...German word for a mathematical field) is a [[set]] of elements with two [[operation]]s, usually called multiplication and addition (denoted <math>\cdot</math>
  2 KB (362 words) - 23:24, 31 December 2021
• Group
  ...f <math>a\cdot b</math>) satisfying the following conditions, known as the group axioms: ...math> so that <math>gg^{-1}=g^{-1}g=e</math> ([[Inverse with respect to an operation | inverses]]).
  2 KB (365 words) - 12:03, 12 November 2023
• Inverse
  * [[Inverse with respect to an operation]], such as in a [[group]] (see also [[identity]])
  334 bytes (52 words) - 11:42, 23 November 2007
• Word problem
  Word problems often have phrases that indicate which math operation to use. For instance, the phrases "in total" and "altogether" are likely a ...ing the results. For instance, when determining the number of members per group when dividing 19 students into 4 teams, we shouldn't say that each team wou
  1 KB (200 words) - 15:05, 10 April 2020
• Vector space
  ...denoted <math>\mathbf 0</math>) and additive [[inverse with respect to an operation | inverses]].
  3 KB (561 words) - 00:47, 21 March 2009
• Noncommutative
  More formally, if <math>\star</math> is some [[binary operation]] on a [[set]], and <math>x</math> and <math>y</math> are elements of that ...ies of a regular n-gon form a noncommutative [[group]] called a [[dihedral group]].
  2 KB (257 words) - 15:30, 26 December 2017
• Abelian group
  An '''abelian group''' is a [[group]] in which the group [[operation]] is [[commutative]]. They are named after Norwegian mathematician Niels Ab For a [[group]] to be considered '''abelian''', it must meet several requirements.
  2 KB (346 words) - 18:30, 14 June 2020
• Free group
  ...> is the group [[identity]] and is equal to the empty string. The group [[operation]] is concatenation. An example of an element of the free group on <math>I = \{1, 2\}</math> is <math>X_1X_2^{-1}X_1^{-1}X_2^3</math> (wher
  2 KB (454 words) - 17:54, 16 March 2012
• Multiplicative function
  ...with this property are [[homomorphism]]s of [[group]]s (where the [[group operation]] is multiplication).
  3 KB (450 words) - 12:59, 21 July 2009
• Monoid
  A '''monoid''' is a set <math>S</math> closed under an [[operation]] <math>\times</math> which is defined everywhere on <math>S</math>, is [[a ...noid can be thought of as a [[group]] without [[inverse with respect to an operation | inverses]], or as an associative [[magma]] with an identity.
  3 KB (670 words) - 22:45, 21 May 2008
• Closure
  ...is a property of an [[abstract algebra]]ic structure, such as a [[set]], [[group]], [[ring]], or [[field]] ...ds, when any two members of <math>\mathbb{S}</math> are combined using the operation, the result also is a member of <math>\mathbb{S}</math>.
  1 KB (208 words) - 21:55, 20 August 2008
• Symmetric group
  ...[function]] from a [[set]] of size <math>n</math> to itself, and the group operation is [[composition]] of functions. ...or example, an important theorem in [[Galois theory]] is that the [[Galois group]] of the general polynomial equation of degree <math>n</math> is <math>S_{n
  10 KB (1,668 words) - 15:33, 25 May 2008
• 2008 USAMO Problems/Problem 6
  === Solution 2 (group theory) === ...0, 0, \ldots, 0)</math>, is in this set. We claim this set is an [[abelian group]] under [[composition]].
  13 KB (2,414 words) - 14:37, 11 July 2016
• Zassenhaus's Lemma
  ...a doctorate student under Emil Artin at the time. In this article, group operation is written multiplicatively. ...subgroup of <math>K' \cdot (H \cap K)</math>; furthermore, the [[quotient group]]s
  2 KB (414 words) - 12:13, 9 April 2019
• Homogeneous set
  ...] on a set <math>S</math>. If <math>S</math> has only one orbit, then the operation of <math>G</math> on <math>S</math> is said to be ''transitive'', and the < ...n each of the [[orbit]]s of <math>S</math> is homogenous under the induced operation of <math>G</math>.
  7 KB (1,332 words) - 18:45, 9 September 2008

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