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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>.
  7 KB (1,332 words) - 18:45, 9 September 2008
• 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<
  3 KB (524 words) - 13:28, 21 February 2017
• 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]]
  3 KB (616 words) - 22:13, 12 January 2017
• 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>
  3 KB (488 words) - 21:51, 27 May 2008
• 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]].
  11 KB (2,071 words) - 12:25, 9 April 2019
• 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.
  2 KB (278 words) - 12:53, 6 June 2008
• 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.
  5 KB (792 words) - 19:01, 7 April 2012
• 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.
  883 bytes (156 words) - 20:11, 23 January 2017
• 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
  8 KB (1,518 words) - 20:11, 23 January 2017
• 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?
  15 KB (2,297 words) - 12:57, 19 February 2020
• 2014 USAMO Problems/Problem 3
  ==Solution (Group Theory)== ...of these points need to be point at infinity (the identity element of the group).
  2 KB (451 words) - 19:09, 1 May 2014
• 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
  18 KB (2,788 words) - 13:55, 20 February 2020
• 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.
  2 KB (365 words) - 03:04, 26 August 2017
• 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
  13 KB (2,059 words) - 02:59, 21 January 2021
• 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

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