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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
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• 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
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• 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
• 1986 AIME Problems/Problem 13
  ...each of the sequences of two coin tosses as an [[operation]] instead; this operation takes a string and adds the next coin toss on (eg, <tt>THHTH</tt> + <tt>HT< ...the first four spaces, we gain back another TT sequence. We start with one group that has four TT sequences, so when we place them in the spaces, we get <ma
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• 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]]
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• 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]];
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• 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>
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• 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]]).
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• Inverse
  * [[Inverse with respect to an operation]], such as in a [[group]] (see also [[identity]])
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• 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
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• Vector space
  ...denoted <math>\mathbf 0</math>) and additive [[inverse with respect to an operation | inverses]].
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• 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.
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• 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.
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• 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>.
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• 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]].
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• 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
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• 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>.
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• Homogeneous principal set
  A '''homogenous principal set''' is a type of [[group]] [[group action|action]] on a [[set]]. ...a group with a left operation on a set <math>S</math>. The <math>G</math>-group <math>S</math> is called a '''left homogeneous principal set under <math>G<
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• Cycle (permutation)
  ...h> is a '''cycle''' if <math>M</math> has exactly one [[orbit]] (under the operation of <math>\bar{\zeta}</math>) which does not consist of a single [[element]] * [[Symmetric group]]
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• Semi-direct product
  The '''(external) semi-direct''' product, in [[group theory]], is a generalization of the [[direct product]]. Let <math>E</math> be a group, <math>F</math> a [[normal subgroup]] of <math>E</math>, and <math>G</math>
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• Sylow Theorems
  ...ylow theorems''' are a collection of results in the theory of [[finite]] [[group]]s. They give a partial converse to [[Lagrange's Theorem]], and are one of :'''Theorem.''' Every finite group contains a [[Sylow p-subgroup |Sylow <math>p</math>-subgroup]].
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• Direct sum
  ...ructures with [[identity]] elements. The construction can be applied to [[group]]s, [[ring]]s, and [[module]]s. ...an group]]s, since additive notation is usually used for a [[commutative]] operation.
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• Category (category theory)
  * For every three objects, <math>A,B,C \in \mathcal{C}</math>, a binary operation <math>\circ: \text{Hom}(B,C) \times \text{Hom}(A,B) \to \text{Hom}(A,C)</ma ...etely abstractly (similarly to how we study multiplication abstractly in [[group theory]]), and never talk about 'plugging things in to' morphisms.
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• Module
  ...ath>-module''' is an [[abelian group]] <math>(M,+)</math> together with an operation <math>R\times M\to M</math> (called scalar multiplication) written as <math ...write <math>M</math> to mean the module as well as the underlying abelian group.
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• AoPS Wiki:Competition ratings
  ...plit this collection into <math>100</math> or fewer groups, such that each group has total value at most <math>1</math>. ..._1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <ma
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• Fundamental group
  ...s the simplest object of study in algebraic topology is the '''fundamental group'''. Now define a [[binary operation]] <math>\cdot</math> (called ''concatenation'') on <math>\Omega(X,x_0)</mat
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• 2007 AMC 10B Problems
  Define the operation <math>\star</math> by <math>a \star b = (a+b)b.</math> What is <math>(3 \st ...h>30\%</math> of the group are girls. How many girls were initially in the group?
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• 2014 USAMO Problems/Problem 3
  ==Solution (Group Theory)== ...of these points need to be point at infinity (the identity element of the group).
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• 1973 AHSME Problems
  Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the
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• 1971 AHSME Problems
  Let <math>\ast</math> be the symbol denoting the binary operation on the set <math>S</math> of all non-zero real numbers as follows: Each of a group of <math>50</math> girls is blonde or brunette and is blue eyed of brown ey
  18 KB (2,703 words) - 20:50, 11 September 2023
• 2014 IMO Problems/Problem 5
  ...plit this collection into <math>100</math> or fewer groups, such that each group has total value at most <math>1</math>. ...ac{1}{2m+1}</math> appears <math>2m+1</math> times, group it into a single group and induct downwards.
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• 2015 USAJMO Problems/Problem 1
  ...ve integers from <math>-1007</math> to <math>1007</math>. Notice that the operation we are applying in this problem does not change the sum or the mean of the ...1}, m_{111})</math> for all the eight elements. Since the sum of the eight-group is <math>0</math>, <math>m_{111}</math> must also be <math>0</math>. Theref
  8 KB (1,405 words) - 20:13, 26 July 2022
• Mock AMC 10B Problems
  ...is a positive integer greater than 2 and <math>n < \mu</math>. After this operation, the line segments <math>A_0A_1</math>, <math>A_1A_2</math>, <math>A_2A_3</ ...ath>32</math> students. To split the class up into partners that work on a group project involving integrals, the teacher, Mrs. Jannesen, randomly partition
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• 2019 IMO Problems/Problem 5
  ...rranged in a line from left to right. He repeatedly performs the following operation: Proof: Let us have <math>k \ge 1</math> heads in our group of n coins, of which <math>k-1</math> are in the first n-1. We are supposed
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• User:Vqbc/Testing2
  ...ple each person shakes hands with exactly two of the other people from the group. Let <math>N</math> be the number of ways this handshaking can occur. Consi ...oups (some of which may be empty) such that the sum of the numbers in each group is at most <math>1</math>.
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• User:Smeymage
  ...functioning. In spite of this, they usually are misused that will create a group of uncomfortable side effects. They've also been legal to make usage of doc ...e health conditions. They're just typically driven to improve the tangible operation of people who require a malady identified as physical body dysmorphic disor
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• User:Lyeismos
  ...cts are one of the most important features of JavaScript. They allow us to group together data and functionality into a single unit that can be easily worke ...These tasks can be as simple as returning a boolean value or performing an operation on an array.
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• User:Firadroth
  ...irst scheduling and design stages to your closing add-ons about build, the group of industry experts is equipped to cope with every part of building your pr ...customers, insuring transparency and also union all through the framework operation. Their own team of faithful specialists runs faithfully utilizing clients,
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• User talk:Iloveminersminecraftshirt
  ...Surface mining poses its own set of challenges, including heavy machinery operation and exposure to extreme weather conditions. ..., while the word “miners” proudly showcases support for this dedicated group of individuals.
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