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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
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• 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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• 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
  10 KB (1,760 words) - 01:51, 19 November 2023