Definition of a group Lesson 24
In this video we take our first look at the definition of a group. It is basically a set of elements and the operation defined on them. If this set of elements and the operation defined on them obey the properties of closure and associativity, and if one of the elements is the identity el
From playlist Abstract algebra
Symmetric Groups (Abstract Algebra)
Symmetric groups are some of the most essential types of finite groups. A symmetric group is the group of permutations on a set. The group of permutations on a set of n-elements is denoted S_n. Symmetric groups capture the history of abstract algebra, provide a wide range of examples in
From playlist Abstract Algebra
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
Visual Group Theory, Lecture 1.6: The formal definition of a group
Visual Group Theory, Lecture 1.6: The formal definition of a group At last, after five lectures of building up our intuition of groups and numerous examples, we are ready to present the formal definition of a group. We conclude by proving several basic properties that are not built into t
From playlist Visual Group Theory
Abstract Algebra | Cyclic Subgroups
We define the notion of a cyclic subgroup and give a few examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
What is a Group? | Abstract Algebra
Welcome to group theory! In today's lesson we'll be going over the definition of a group. We'll see the four group axioms in action with some examples, and some non-examples as well which violate the axioms and are thus not groups. In a fundamental way, groups are structures built from s
From playlist Abstract Algebra
Cyclic Groups (Abstract Algebra)
Cyclic groups are the building blocks of abelian groups. There are finite and infinite cyclic groups. In this video we will define cyclic groups, give a list of all cyclic groups, talk about the name “cyclic,” and see why they are so essential in abstract algebra. Be sure to subscribe s
From playlist Abstract Algebra
Group Theory: The Center of a Group G is a Subgroup of G Proof
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Group Theory: The Center of a Group G is a Subgroup of G Proof
From playlist Abstract Algebra
Abstract Algebra | Subgroups of Cyclic Groups
We prove that all subgroups of cyclic groups are themselves cyclic. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
Strange Pattern in symmetries - Bott periodicity
A strange repeating pattern in the symmetries of circles, spheres and higher dimensional spheres called Bott periodicity. We will learn about symmetries of spheres, homotopy groups, the orthogonal groups and, finally, Bott periodicity. Produced by Connect films https://www.connectfi
From playlist Summer of Math Exposition Youtube Videos
Lie Groups and Lie Algebras: Lesson 35 - The Fundamental Group
Lie Groups and Lie Algebras: Lesson 35 - The Fundamental Group Now that we understand the notion of homotopic paths ina topological space, we focus on loops. Using the fact that homotopy is an equivalence relation we create a set of equivalence classes of homotopic loops. That set is give
From playlist Lie Groups and Lie Algebras
Distiguished Lecture Series by Richard Melrose (Massachusetts Institute of Technology, USA)
From playlist Distinguished Visitors Lecture Series
Lie Groups and Lie Algebras: Lesson 37 - The Fundamental Groups of SU(2) and SO(3)
Lie Groups and Lie Algebras: Lesson 37 - Homotopy Groups of SU(2) and SO(3) In this lesson we discover the Fundamental Group of SU(2) and S0(3) and learn the critical fact that they are not the same. That is, the Fundamental Group associated with the topological space SU(2) is simply conn
From playlist Lie Groups and Lie Algebras
Algebraic topology: Fundamental group
This lecture is part of an online course on algebraic topology. We define the fundamental group, calculate it for some easy examples (vector spaces and spheres), and give a couple of applications (R^2 is not homeomorphic to R^3, the Brouwer fixedpoint theorem). For the other lectures in
From playlist Algebraic topology
In this video, we introduce and discuss spectra (in the sense of homotopy theory). We explain how they generalise abelian groups. Feel free to post comments and questions at our public forum at https://www.uni-muenster.de/TopologyQA/index.php?qa=tc-lecture Homepage with further informa
From playlist Higher Algebra
AlgTop25: More on the fundamental group
This video continues our discussion of the fundamental group of a space. We show that the homotopy classes of closed loops from a fixed point on a space actually form a group. And the important cases of the torus and the projective plane are studied in some detail. This is the 25th lect
From playlist Algebraic Topology: a beginner's course - N J Wildberger
Algebraic Topology 1.4 : Fundamental Group
In this video, I introduce the fundamental group, and explain the induced isomorphism resulting from a path and the induced homomorphism resulting from a continuous map, proving functorality. I also briefly cover retractions and how their induced homomorphism is surjective. Translate This
From playlist Topology
Group Definition (expanded) - Abstract Algebra
The group is the most fundamental object you will study in abstract algebra. Groups generalize a wide variety of mathematical sets: the integers, symmetries of shapes, modular arithmetic, NxM matrices, and much more. After learning about groups in detail, you will then be ready to contin
From playlist Abstract Algebra
Ulrike Tillmann, Lecture II - 11 February 2015
Ulrike Tillmann (University of Oxford) - Lecture II http://www.crm.sns.it/course/4038/ Mapping class groups and diffeomorphism groups of manifolds play an important role in geometry and topology. We will discuss recent advances in the understanding of their homology exploring homotopy the
From playlist Algebraic topology, geometric and combinatorial group theory - 2015