Quotient objects | Group theory
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written , where is the original group and is the normal subgroup. (This is pronounced , where is short for modulo.) Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of . Specifically, the image of under a homomorphism is isomorphic to where denotes the kernel of . The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.(For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set.) (Wikipedia).
Now that we know what a quotient group is, let's take a look at an example to cement our understanding of the concepts involved.
From playlist Abstract algebra
The idea of a quotient group follows easily from cosets and Lagrange's theorem. In this video, we start with a normal subgroup and develop the idea of a quotient group, by viewing each coset (together with the normal subgroup) as individual mathematical objects in a set. This set, under
From playlist Abstract algebra
Visual Group Theory, Lecture 3.5: Quotient groups
Visual Group Theory, Lecture 3.5: Quotient groups Like how a direct product can be thought of as a way to "multiply" two groups, a quotient is a way to "divide" a group by one of its subgroups. We start by defining this in terms of collapsing Cayley diagrams, until we get a conjecture abo
From playlist Visual Group Theory
Chapter 5: Quotient groups | Essence of Group Theory
Quotient groups is a very important concept in group theory, because it has paramount importance in group homomorphisms (connection with the isomorphism theorem(s)). With this video series, abstract algebra needs not be abstract - one can easily develop intuitions for group theory! In fac
From playlist Essence of Group Theory
Abstract Algebra | Quotient Groups
We introduce the notion of a quotient group and give some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
Now that we have defined and understand quotient groups, we need to look at product groups. In this video I define the product of two groups as well as the group operation, proving that it is indeed a group.
From playlist Abstract algebra
Every Group is a Quotient of a Free Group
First isomorphism theorem: https://youtu.be/ssVIJO5uNeg An explanation of a proof that every finite group is a quotient of a free group. A similar proof also applies to infinite groups because we can consider a free group on an infinite number of elements! Group Theory playlist: https://
From playlist Group Theory
Why Normal Subgroups are Necessary for Quotient Groups
Proof that cosets are disjoint: https://youtu.be/uxhAUmgSHnI In order for a subgroup to create a quotient group (also known as factor group), it must be a normal subgroup. That means that when we conjugate an element in the subgroup, it stays in the subgroup. In this video, we explain wh
From playlist Group Theory
RNT1.4. Ideals and Quotient Rings
Ring Theory: We define ideals in rings as an analogue of normal subgroups in group theory. We give a correspondence between (two-sided) ideals and kernels of homomorphisms using quotient rings. We also state the First Isomorphism Theorem for Rings and give examples.
From playlist Abstract Algebra
Prayagdeep Parija: Random Quotients of Hyperbolic Groups and Property (T)
Prayagdeep Parija, University of Wisconsin Milwaukee Title: Random Quotients of Hyperbolic Groups and Property (T) What does a typical quotient of a group look like? Gromov had looked at density model of quotients of free groups. The density parameter d measures the rate of exponential gro
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Martin Bridson - Profinite isomorphism problems.
Martin Bridson (University of Oxford, England)
From playlist T1-2014 : Random walks and asymptopic geometry of groups.
Profinite rigidity – Alan Reid – ICM2018
Topology Invited Lecture 6.7 Profinite rigidity Alan Reid Abstract: We survey recent work on profinite rigidity of residually finite groups. © International Congress of Mathematicians – ICM www.icm2018.org Os direitos sobre todo o material deste canal pertencem ao Instituto de Mat
From playlist Topology
Profinite rigidity and flexibility for compact 3-manifold groups -Reid
Geometric Structures on 3-manifolds Topic:Profinite rigidity and flexibility for compact 3-manifold groups Speaker: Alan Reid Date: Tuesday, February 2 This talk will discuss the question: To what extent are the fundamental groups of compact 3-manifolds determined (amongst the fundamental
From playlist Mathematics
Visual Group Theory, Lecture 4.5: The isomorphism theorems
Visual Group Theory, Lecture 4.5: The isomorphism theorems There are four central results in group theory that are collectively known at the isomorphism theorems. We introduced the first of these a few lectures back, under the name of the "fundamental homomorphism theorem." In this lectur
From playlist Visual Group Theory
A New Cubulation Theorem for Hyperbolic Groups- Daniel Groves
Daniel Groves University of Illinois, Chicago October 27, 2015 https://www.math.ias.edu/seminars/abstract?event=83384 We prove that if a hyperbolic group G acts cocompactly on a CAT(0) cube complexes and the cell stabilizers are quasiconvex and virtually special, then G is virtually spec
From playlist Geometric Structures on 3-manifolds
Group theory 29:The Jordan Holder theorem
This lecture is part of an online course on group theory. It covers the Jordan-Holder theorem, staring that the simple groups appearing in a composition series of a finite group do not depend on the composition series.
From playlist Group theory
Grothendieck Pairs and Profinite Rigidity - Martin Bridson
Arithmetic Groups Topic: Grothendieck Pairs and Profinite Rigidity Speaker: Martin Bridson Affiliation: Oxford University Date: January 26, 2022 If a monomorphism of abstract groups H↪G induces an isomorphism of profinite completions, then (G,H) is called a Grothendieck pair, recalling t
From playlist Mathematics
Finding the Quotient Group Example with Integers
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Finding the Quotient Group Example with Integers
From playlist Abstract Algebra
Parahoric Subgroups and Supercuspidal Representations of p-Adic groups - Dick Gross
Dick Gross Harvard University December 9, 2010 This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is th
From playlist Mathematics