- published: 28 Jun 2020
- views: 32774
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Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.
In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.
A group action is an extension to the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Group_action
Podcasts:
-
Chapter 7: Group actions, symmetric group and Cayley’s theorem | Essence of Group Theory
Group action can be thought of as a homomorphism to a symmetric group, so apart from orbit-stabiliser theorem, we can also use the isomorphism theorem to analyse any group action. It turns out that this correspondence between group action and homomorphism can be visualised rather easily. This correspondence is very important in group theory, but often neglected. This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. Symmetric group is also briefly mentioned here as a concept to ...
published: 28 Jun 2020 -
Group Theory, lecture 5.1: Group actions
We discuss group actions
published: 15 Nov 2020 -
Chapter 1: Symmetries, Groups and Actions | Essence of Group Theory
Start of a video series on intuitions of group theory. Groups are often introduced as a kind of abstract algebraic object right from the start, which is not good for developing intuitions for first-time learners. This video series hopes to help you develop intuitions, which are useful in understanding group theory. In particular, this video is going to be about thinking groups as symmetries (or isometries to be precise) because they are much more visualisable, and that symmetries of an object do form a group using the abstract definition of the group that is usually given. Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to kn...
published: 06 Feb 2020 -
Group Theory: Lecture 15/30 - Introduction to Group Actions
This video series is not endorsed by the University of Cambridge. These videos are primarily inspired from Dexter Chua's lecture notes, which can be found here: https://dec41.user.srcf.net/notes/IA_M/groups.pdf Additionally, problem sets for this video series can be found here: https://dec41.user.srcf.net/notes/IA_M/groups_eg.pdf
published: 26 Jun 2023 -
Lecture 33 - Group actions
published: 06 May 2019 -
Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory
An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerges very apparently when counting the total number of symmetries in some tricky but easy way. This video series continues to develop your intuition towards some fundamental concepts and results in Group theory. This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better video...
published: 17 Feb 2020 -
GT15. Group Actions
Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, which in turn yields a partition of X into orbits. Orbits are also described as cosets of the group. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html
published: 23 Jan 2012 -
Group Actions
published: 01 Dec 2020 -
What is a Group Action? : A Group as a Category and The Skeleton Operation ☠
This week I try to take a more Categorical approach to answering and expanding upon the question of "what is a group action". Along the way I'll go over thinking about a group as a category and eventually hit on the skeleton operation on a category and use it to present an example of the categorification of the Orbit-Stabilizer theorem. Here are some videos that are "pre-reqs", that is, they introduce some topics in a bit more detail than I do here. What are groups: https://youtu.be/uraPtd0yCI4 What are categories: https://youtu.be/ZaLB0MF2TMQ Also, Here is the Category Theory Book that inspired this video: https://amzn.to/317hxMc CORRECTIONS: 1. When introducing Groups, while glossing over the group axioms I forget to mention that a group, G, must be closed under the binary operation. ...
published: 09 Jul 2020 -
Group Actions: Stabilizers
Introducing the stabilizer subgroup. Setup for Sylow's Theorem. For more math, subscribe to my channel: https://www.youtube.com/jeffsuzuki1
published: 20 Apr 2022
developed with YouTube
10:51
Chapter 7: Group actions, symmetric group and Cayley’s theorem | Essence of Group Theory
- Order: Reorder
- Duration: 10:51
- Uploaded Date: 28 Jun 2020
- views: 32774
Group action can be thought of as a homomorphism to a symmetric group, so apart from orbit-stabiliser theorem, we can also use the isomorphism theorem to analys...
Group action can be thought of as a homomorphism to a symmetric group, so apart from orbit-stabiliser theorem, we can also use the isomorphism theorem to analyse any group action. It turns out that this correspondence between group action and homomorphism can be visualised rather easily. This correspondence is very important in group theory, but often neglected.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
Symmetric group is also briefly mentioned here as a concept to facilitate the introduction of the above correspondence, but a more detailed analysis of the symmetric group would not be in this video series, because I don't think it is as intuitive as concepts discussed in the video series and therefore does not fit the theme of the series too well. However, I will do a video on Burnside's lemma and its interesting application, but it is not "Essence of Group Theory" anymore, because it is an application of the orbit-stabiliser theorem. I haven't mentioned in the video that I will also not do a video on matrix groups because it requires the knowledge of linear algebra, and again, the concepts discussed in the matrix groups will usually be algebraic.
The next few videos will be a summary of this video series with one very cool example that really covers all the concepts discussed in this video series; another video on the current epidemic explaining an alternative model to the SIR discussed on this channel before, and will be about a stochastic branching process; then the Burnside's lemma video. There might be some videos in between these, but I will definitely do all of these videos in some time in the future.
Cayley's theorem is named in honour of Arthur Cayley, a British (Cambridge) mathematician who is also known for a lot of mathematical results, like Cayley table, Cayley graph, Cayley's theorem and the famous Cayley-Hamilton theorem in linear algebra. However, even though it is a motivation for the study of symmetric groups, you might not see Cayley's theorem too often in more advanced studies of group theory. It mainly acts as a direct application of the correspondence discussed in this video.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #grouptheory #groupaction #homomorphism #Cayley
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
https://wn.com/Chapter_7_Group_Actions,_Symmetric_Group_And_Cayley’S_Theorem_|_Essence_Of_Group_Theory
Group action can be thought of as a homomorphism to a symmetric group, so apart from orbit-stabiliser theorem, we can also use the isomorphism theorem to analyse any group action. It turns out that this correspondence between group action and homomorphism can be visualised rather easily. This correspondence is very important in group theory, but often neglected.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
Symmetric group is also briefly mentioned here as a concept to facilitate the introduction of the above correspondence, but a more detailed analysis of the symmetric group would not be in this video series, because I don't think it is as intuitive as concepts discussed in the video series and therefore does not fit the theme of the series too well. However, I will do a video on Burnside's lemma and its interesting application, but it is not "Essence of Group Theory" anymore, because it is an application of the orbit-stabiliser theorem. I haven't mentioned in the video that I will also not do a video on matrix groups because it requires the knowledge of linear algebra, and again, the concepts discussed in the matrix groups will usually be algebraic.
The next few videos will be a summary of this video series with one very cool example that really covers all the concepts discussed in this video series; another video on the current epidemic explaining an alternative model to the SIR discussed on this channel before, and will be about a stochastic branching process; then the Burnside's lemma video. There might be some videos in between these, but I will definitely do all of these videos in some time in the future.
Cayley's theorem is named in honour of Arthur Cayley, a British (Cambridge) mathematician who is also known for a lot of mathematical results, like Cayley table, Cayley graph, Cayley's theorem and the famous Cayley-Hamilton theorem in linear algebra. However, even though it is a motivation for the study of symmetric groups, you might not see Cayley's theorem too often in more advanced studies of group theory. It mainly acts as a direct application of the correspondence discussed in this video.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #grouptheory #groupaction #homomorphism #Cayley
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
23:42
Group Theory, lecture 5.1: Group actions
- Order: Reorder
- Duration: 23:42
- Uploaded Date: 15 Nov 2020
- views: 7660
We discuss group actions
We discuss group actions
https://wn.com/Group_Theory,_Lecture_5.1_Group_Actions
We discuss group actions
- published: 15 Nov 2020
- views: 7660
6:07
Chapter 1: Symmetries, Groups and Actions | Essence of Group Theory
- Order: Reorder
- Duration: 6:07
- Uploaded Date: 06 Feb 2020
- views: 118551
Start of a video series on intuitions of group theory. Groups are often introduced as a kind of abstract algebraic object right from the start, which is not goo...
Start of a video series on intuitions of group theory. Groups are often introduced as a kind of abstract algebraic object right from the start, which is not good for developing intuitions for first-time learners. This video series hopes to help you develop intuitions, which are useful in understanding group theory.
In particular, this video is going to be about thinking groups as symmetries (or isometries to be precise) because they are much more visualisable, and that symmetries of an object do form a group using the abstract definition of the group that is usually given.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #grouptheory #groups #intuition
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
For my contact email, check my About page on a PC.
See you next time!
https://wn.com/Chapter_1_Symmetries,_Groups_And_Actions_|_Essence_Of_Group_Theory
Start of a video series on intuitions of group theory. Groups are often introduced as a kind of abstract algebraic object right from the start, which is not good for developing intuitions for first-time learners. This video series hopes to help you develop intuitions, which are useful in understanding group theory.
In particular, this video is going to be about thinking groups as symmetries (or isometries to be precise) because they are much more visualisable, and that symmetries of an object do form a group using the abstract definition of the group that is usually given.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #grouptheory #groups #intuition
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
For my contact email, check my About page on a PC.
See you next time!
- published: 06 Feb 2020
- views: 118551
48:59
Group Theory: Lecture 15/30 - Introduction to Group Actions
- Order: Reorder
- Duration: 48:59
- Uploaded Date: 26 Jun 2023
- views: 1044
This video series is not endorsed by the University of Cambridge. These videos are primarily inspired from Dexter Chua's lecture notes, which can be found here:...
This video series is not endorsed by the University of Cambridge. These videos are primarily inspired from Dexter Chua's lecture notes, which can be found here:
https://dec41.user.srcf.net/notes/IA_M/groups.pdf
Additionally, problem sets for this video series can be found here:
https://dec41.user.srcf.net/notes/IA_M/groups_eg.pdf
https://wn.com/Group_Theory_Lecture_15_30_Introduction_To_Group_Actions
This video series is not endorsed by the University of Cambridge. These videos are primarily inspired from Dexter Chua's lecture notes, which can be found here:
https://dec41.user.srcf.net/notes/IA_M/groups.pdf
Additionally, problem sets for this video series can be found here:
https://dec41.user.srcf.net/notes/IA_M/groups_eg.pdf
- published: 26 Jun 2023
- views: 1044
24:53
Lecture 33 - Group actions
- Order: Reorder
- Duration: 24:53
- Uploaded Date: 06 May 2019
- views: 41260
- published: 06 May 2019
- views: 41260
12:27
Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory
- Order: Reorder
- Duration: 12:27
- Uploaded Date: 17 Feb 2020
- views: 68463
An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerges very apparently when counting the total number of symmetries in som...
An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerges very apparently when counting the total number of symmetries in some tricky but easy way. This video series continues to develop your intuition towards some fundamental concepts and results in Group theory.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
#mathemaniac #math #grouptheory #groups #orbitstabiliser #orbitstabilizer
https://wn.com/Chapter_2_Orbit_Stabiliser_Theorem_|_Essence_Of_Group_Theory
An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerges very apparently when counting the total number of symmetries in some tricky but easy way. This video series continues to develop your intuition towards some fundamental concepts and results in Group theory.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
#mathemaniac #math #grouptheory #groups #orbitstabiliser #orbitstabilizer
- published: 17 Feb 2020
- views: 68463
20:19
GT15. Group Actions
- Order: Reorder
- Duration: 20:19
- Uploaded Date: 23 Jan 2012
- views: 45286
Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, ...
Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, which in turn yields a partition of X into orbits. Orbits are also described as cosets of the group.
U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html
https://wn.com/Gt15._Group_Actions
Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, which in turn yields a partition of X into orbits. Orbits are also described as cosets of the group.
U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html
- published: 23 Jan 2012
- views: 45286
41:51
Group Actions
- Order: Reorder
- Duration: 41:51
- Uploaded Date: 01 Dec 2020
- views: 7360
- published: 01 Dec 2020
- views: 7360
20:53
What is a Group Action? : A Group as a Category and The Skeleton Operation ☠
- Order: Reorder
- Duration: 20:53
- Uploaded Date: 09 Jul 2020
- views: 3870
This week I try to take a more Categorical approach to answering and expanding upon the question of "what is a group action". Along the way I'll go over thinkin...
This week I try to take a more Categorical approach to answering and expanding upon the question of "what is a group action". Along the way I'll go over thinking about a group as a category and eventually hit on the skeleton operation on a category and use it to present an example of the categorification of the Orbit-Stabilizer theorem. Here are some videos that are "pre-reqs", that is, they introduce some topics in a bit more detail than I do here.
What are groups: https://youtu.be/uraPtd0yCI4
What are categories: https://youtu.be/ZaLB0MF2TMQ
Also, Here is the Category Theory Book that inspired this video: https://amzn.to/317hxMc
CORRECTIONS:
1. When introducing Groups, while glossing over the group axioms I forget to mention that a group, G, must be closed under the binary operation. That is for any a,b in G, ab must also be in G.
This video can be broken up into the following sections.
00:00 Intro
I once again fail to say "Hi I'm Nathan and Welcome to my YouTube Channel" but I still introduce the goal of the video lol.
00:32 What is a Group Action?
I briefly revisit the idea of what a group is before "complicating" it into the object that we'll spend most of the time interacting with, a group action. There are many different types of group actions but we will focus on 2 examples.
03:46 Example 1
The first of 2 group action examples. We look at probably the most accessible example of a group action, a group acting on itself with the action operating the same way as the group operation.
05:05 Example 2
In the second of 2 group action examples we look at another small but more complicated example of a group acting on itself, here we us the conjugate action and we'll continue to look at this example throughout the video.
After the second example, the video begins to bring in a lot more ideas from category theory and we'll spend a lot of time looking at categories and groups and how the two can correspond to one another.
09:30 A Group as a Category
Here I will begin translating the idea of what we have talked about with groups and group actions into a more category theoretic context. We won't dive too deep into the Category of G-Sets, but we do define what G-Sets are and how they can be described as functors.
11:21 The Translation Groupoid of A Group
Next we 'zoom in' on what our G-Set functor does to the categorical group and arrive at the directed graphs that we looked at previously by looking at the Translation Groupoid.
13:27 The Skeleton Operation
The last Category theoretic tool we will need for the categorification of the orbit-stabilizer theorem, the skeleton operator on a category fuses together identical (or isomorphic) objects, which allows us to see the information in the Translation Groupoid category more clearly.
15:50 The Orbit-Stabilizer Theorem
Here we introduce (not prove...) the theorem and then talk through how the skeleton operation on the translation groupoid generates a very nice picture of the theorem in a Category Theoretic context.
17:35 Another way of Thinking about the Skeleton Category
And lastly I pull away from the abstract nonsense of the skeleton construction to quickly walk through another way to interpret each of the orbits in the skeleton category that can help when trying to determine the size of other symmetry/dihedral groups.
_____________________
Last PhD Update: https://youtu.be/de_MVF1FiZU
Last Time: https://youtu.be/MjyBP7m12yg
The CHALKboard: http://www.youtube.com/c/CHALKboard
Find the CHALKboard on Facebook: http://bit.ly/CHALKboard
_____________________
WHAT GEAR I USED FOR THIS VIDEO:
Camera - https://amzn.to/30bJRho
Lens - https://amzn.to/3fgDaPa
SD card - https://amzn.to/2PbTiag
Microphone - https://amzn.to/2D3KYXE
Lighting Kit - https://amzn.to/3ffibMH
Editing Hard Drive - https://amzn.to/2P7hSsM
DISCLAIMER: Links included in this description may be affiliate links. If you purchase a product or service with the links that I include here, I may receive a small commission. There is no additional charge to you! Thank you for supporting my channel and for helping me create these free videos!
Subscriber Count as of release: 1,783
#CHALK #CategoryTheory #Skeleton
https://wn.com/What_Is_A_Group_Action_A_Group_As_A_Category_And_The_Skeleton_Operation_☠
This week I try to take a more Categorical approach to answering and expanding upon the question of "what is a group action". Along the way I'll go over thinking about a group as a category and eventually hit on the skeleton operation on a category and use it to present an example of the categorification of the Orbit-Stabilizer theorem. Here are some videos that are "pre-reqs", that is, they introduce some topics in a bit more detail than I do here.
What are groups: https://youtu.be/uraPtd0yCI4
What are categories: https://youtu.be/ZaLB0MF2TMQ
Also, Here is the Category Theory Book that inspired this video: https://amzn.to/317hxMc
CORRECTIONS:
1. When introducing Groups, while glossing over the group axioms I forget to mention that a group, G, must be closed under the binary operation. That is for any a,b in G, ab must also be in G.
This video can be broken up into the following sections.
00:00 Intro
I once again fail to say "Hi I'm Nathan and Welcome to my YouTube Channel" but I still introduce the goal of the video lol.
00:32 What is a Group Action?
I briefly revisit the idea of what a group is before "complicating" it into the object that we'll spend most of the time interacting with, a group action. There are many different types of group actions but we will focus on 2 examples.
03:46 Example 1
The first of 2 group action examples. We look at probably the most accessible example of a group action, a group acting on itself with the action operating the same way as the group operation.
05:05 Example 2
In the second of 2 group action examples we look at another small but more complicated example of a group acting on itself, here we us the conjugate action and we'll continue to look at this example throughout the video.
After the second example, the video begins to bring in a lot more ideas from category theory and we'll spend a lot of time looking at categories and groups and how the two can correspond to one another.
09:30 A Group as a Category
Here I will begin translating the idea of what we have talked about with groups and group actions into a more category theoretic context. We won't dive too deep into the Category of G-Sets, but we do define what G-Sets are and how they can be described as functors.
11:21 The Translation Groupoid of A Group
Next we 'zoom in' on what our G-Set functor does to the categorical group and arrive at the directed graphs that we looked at previously by looking at the Translation Groupoid.
13:27 The Skeleton Operation
The last Category theoretic tool we will need for the categorification of the orbit-stabilizer theorem, the skeleton operator on a category fuses together identical (or isomorphic) objects, which allows us to see the information in the Translation Groupoid category more clearly.
15:50 The Orbit-Stabilizer Theorem
Here we introduce (not prove...) the theorem and then talk through how the skeleton operation on the translation groupoid generates a very nice picture of the theorem in a Category Theoretic context.
17:35 Another way of Thinking about the Skeleton Category
And lastly I pull away from the abstract nonsense of the skeleton construction to quickly walk through another way to interpret each of the orbits in the skeleton category that can help when trying to determine the size of other symmetry/dihedral groups.
_____________________
Last PhD Update: https://youtu.be/de_MVF1FiZU
Last Time: https://youtu.be/MjyBP7m12yg
The CHALKboard: http://www.youtube.com/c/CHALKboard
Find the CHALKboard on Facebook: http://bit.ly/CHALKboard
_____________________
WHAT GEAR I USED FOR THIS VIDEO:
Camera - https://amzn.to/30bJRho
Lens - https://amzn.to/3fgDaPa
SD card - https://amzn.to/2PbTiag
Microphone - https://amzn.to/2D3KYXE
Lighting Kit - https://amzn.to/3ffibMH
Editing Hard Drive - https://amzn.to/2P7hSsM
DISCLAIMER: Links included in this description may be affiliate links. If you purchase a product or service with the links that I include here, I may receive a small commission. There is no additional charge to you! Thank you for supporting my channel and for helping me create these free videos!
Subscriber Count as of release: 1,783
#CHALK #CategoryTheory #Skeleton
- published: 09 Jul 2020
- views: 3870
4:37
Group Actions: Stabilizers
- Order: Reorder
- Duration: 4:37
- Uploaded Date: 20 Apr 2022
- views: 244
Introducing the stabilizer subgroup. Setup for Sylow's Theorem.
For more math, subscribe to my channel: https://www.youtube.com/jeffsuzuki1
Introducing the stabilizer subgroup. Setup for Sylow's Theorem.
For more math, subscribe to my channel: https://www.youtube.com/jeffsuzuki1
https://wn.com/Group_Actions_Stabilizers
Introducing the stabilizer subgroup. Setup for Sylow's Theorem.
For more math, subscribe to my channel: https://www.youtube.com/jeffsuzuki1
- published: 20 Apr 2022
- views: 244
10:51
Chapter 7: Group actions, symmetric group and Cayley’s theorem | Essence of Group Theory
Group action can be thought of as a homomorphism to a symmetric group, so apart from orbit...
published: 28 Jun 2020
Chapter 7: Group actions, symmetric group and Cayley’s theorem | Essence of Group Theory
Chapter 7: Group actions, symmetric group and Cayley’s theorem | Essence of Group Theory
- Report rights infringement
- published: 28 Jun 2020
- views: 32774
Group action can be thought of as a homomorphism to a symmetric group, so apart from orbit-stabiliser theorem, we can also use the isomorphism theorem to analyse any group action. It turns out that this correspondence between group action and homomorphism can be visualised rather easily. This correspondence is very important in group theory, but often neglected.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
Symmetric group is also briefly mentioned here as a concept to facilitate the introduction of the above correspondence, but a more detailed analysis of the symmetric group would not be in this video series, because I don't think it is as intuitive as concepts discussed in the video series and therefore does not fit the theme of the series too well. However, I will do a video on Burnside's lemma and its interesting application, but it is not "Essence of Group Theory" anymore, because it is an application of the orbit-stabiliser theorem. I haven't mentioned in the video that I will also not do a video on matrix groups because it requires the knowledge of linear algebra, and again, the concepts discussed in the matrix groups will usually be algebraic.
The next few videos will be a summary of this video series with one very cool example that really covers all the concepts discussed in this video series; another video on the current epidemic explaining an alternative model to the SIR discussed on this channel before, and will be about a stochastic branching process; then the Burnside's lemma video. There might be some videos in between these, but I will definitely do all of these videos in some time in the future.
Cayley's theorem is named in honour of Arthur Cayley, a British (Cambridge) mathematician who is also known for a lot of mathematical results, like Cayley table, Cayley graph, Cayley's theorem and the famous Cayley-Hamilton theorem in linear algebra. However, even though it is a motivation for the study of symmetric groups, you might not see Cayley's theorem too often in more advanced studies of group theory. It mainly acts as a direct application of the correspondence discussed in this video.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #grouptheory #groupaction #homomorphism #Cayley
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23:42
Group Theory, lecture 5.1: Group actions
We discuss group actions
published: 15 Nov 2020
Group Theory, lecture 5.1: Group actions
Group Theory, lecture 5.1: Group actions
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- published: 15 Nov 2020
- views: 7660
We discuss group actions
6:07
Chapter 1: Symmetries, Groups and Actions | Essence of Group Theory
Start of a video series on intuitions of group theory. Groups are often introduced as a ki...
published: 06 Feb 2020
Chapter 1: Symmetries, Groups and Actions | Essence of Group Theory
Chapter 1: Symmetries, Groups and Actions | Essence of Group Theory
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- published: 06 Feb 2020
- views: 118551
Start of a video series on intuitions of group theory. Groups are often introduced as a kind of abstract algebraic object right from the start, which is not good for developing intuitions for first-time learners. This video series hopes to help you develop intuitions, which are useful in understanding group theory.
In particular, this video is going to be about thinking groups as symmetries (or isometries to be precise) because they are much more visualisable, and that symmetries of an object do form a group using the abstract definition of the group that is usually given.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #grouptheory #groups #intuition
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See you next time!
48:59
Group Theory: Lecture 15/30 - Introduction to Group Actions
This video series is not endorsed by the University of Cambridge. These videos are primari...
published: 26 Jun 2023
Group Theory: Lecture 15/30 - Introduction to Group Actions
Group Theory: Lecture 15/30 - Introduction to Group Actions
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- published: 26 Jun 2023
- views: 1044
This video series is not endorsed by the University of Cambridge. These videos are primarily inspired from Dexter Chua's lecture notes, which can be found here:
https://dec41.user.srcf.net/notes/IA_M/groups.pdf
Additionally, problem sets for this video series can be found here:
https://dec41.user.srcf.net/notes/IA_M/groups_eg.pdf
24:53
Lecture 33 - Group actions
published: 06 May 2019
Lecture 33 - Group actions
12:27
Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory
An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerg...
published: 17 Feb 2020
Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory
Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory
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- published: 17 Feb 2020
- views: 68463
An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerges very apparently when counting the total number of symmetries in some tricky but easy way. This video series continues to develop your intuition towards some fundamental concepts and results in Group theory.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
#mathemaniac #math #grouptheory #groups #orbitstabiliser #orbitstabilizer
20:19
GT15. Group Actions
Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetri...
published: 23 Jan 2012
GT15. Group Actions
GT15. Group Actions
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- published: 23 Jan 2012
- views: 45286
Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, which in turn yields a partition of X into orbits. Orbits are also described as cosets of the group.
U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html
20:53
What is a Group Action? : A Group as a Category and The Skeleton Operation ☠
This week I try to take a more Categorical approach to answering and expanding upon the qu...
published: 09 Jul 2020
What is a Group Action? : A Group as a Category and The Skeleton Operation ☠
What is a Group Action? : A Group as a Category and The Skeleton Operation ☠
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- published: 09 Jul 2020
- views: 3870
This week I try to take a more Categorical approach to answering and expanding upon the question of "what is a group action". Along the way I'll go over thinking about a group as a category and eventually hit on the skeleton operation on a category and use it to present an example of the categorification of the Orbit-Stabilizer theorem. Here are some videos that are "pre-reqs", that is, they introduce some topics in a bit more detail than I do here.
What are groups: https://youtu.be/uraPtd0yCI4
What are categories: https://youtu.be/ZaLB0MF2TMQ
Also, Here is the Category Theory Book that inspired this video: https://amzn.to/317hxMc
CORRECTIONS:
1. When introducing Groups, while glossing over the group axioms I forget to mention that a group, G, must be closed under the binary operation. That is for any a,b in G, ab must also be in G.
This video can be broken up into the following sections.
00:00 Intro
I once again fail to say "Hi I'm Nathan and Welcome to my YouTube Channel" but I still introduce the goal of the video lol.
00:32 What is a Group Action?
I briefly revisit the idea of what a group is before "complicating" it into the object that we'll spend most of the time interacting with, a group action. There are many different types of group actions but we will focus on 2 examples.
03:46 Example 1
The first of 2 group action examples. We look at probably the most accessible example of a group action, a group acting on itself with the action operating the same way as the group operation.
05:05 Example 2
In the second of 2 group action examples we look at another small but more complicated example of a group acting on itself, here we us the conjugate action and we'll continue to look at this example throughout the video.
After the second example, the video begins to bring in a lot more ideas from category theory and we'll spend a lot of time looking at categories and groups and how the two can correspond to one another.
09:30 A Group as a Category
Here I will begin translating the idea of what we have talked about with groups and group actions into a more category theoretic context. We won't dive too deep into the Category of G-Sets, but we do define what G-Sets are and how they can be described as functors.
11:21 The Translation Groupoid of A Group
Next we 'zoom in' on what our G-Set functor does to the categorical group and arrive at the directed graphs that we looked at previously by looking at the Translation Groupoid.
13:27 The Skeleton Operation
The last Category theoretic tool we will need for the categorification of the orbit-stabilizer theorem, the skeleton operator on a category fuses together identical (or isomorphic) objects, which allows us to see the information in the Translation Groupoid category more clearly.
15:50 The Orbit-Stabilizer Theorem
Here we introduce (not prove...) the theorem and then talk through how the skeleton operation on the translation groupoid generates a very nice picture of the theorem in a Category Theoretic context.
17:35 Another way of Thinking about the Skeleton Category
And lastly I pull away from the abstract nonsense of the skeleton construction to quickly walk through another way to interpret each of the orbits in the skeleton category that can help when trying to determine the size of other symmetry/dihedral groups.
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4:37
Group Actions: Stabilizers
Introducing the stabilizer subgroup. Setup for Sylow's Theorem.
For more math, subscribe...
published: 20 Apr 2022
Group Actions: Stabilizers
Group Actions: Stabilizers
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- published: 20 Apr 2022
- views: 244
Introducing the stabilizer subgroup. Setup for Sylow's Theorem.
For more math, subscribe to my channel: https://www.youtube.com/jeffsuzuki1
Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.
In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.
A group action is an extension to the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Group_action
Chapter 7: Group actions, symmetric group and Cayl...
Group Theory, lecture 5.1: Group actions...
Chapter 1: Symmetries, Groups and Actions | Essenc...
Group Theory: Lecture 15/30 - Introduction to Grou...
Lecture 33 - Group actions...
Chapter 2: Orbit-Stabiliser Theorem | Essence of G...
GT15. Group Actions...
Group Actions...
What is a Group Action? : A Group as a Category an...
Group Actions: Stabilizers...
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