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Quotient group

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 the group structure. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n 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.

In a quotient of 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 G / N, where G is the original group and N is the normal subgroup. (This is pronounced "G mod N," where "mod" 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 G. Specifically, the image of G under a homomorphism φ: G → H is isomorphic to G / ker(φ) where ker(φ) denotes the kernel of φ.

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This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Quotient_group

Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a powerful formalism for analyzing and comparing different algebraic structures.

Universal algebra is a related subject that studies the nature and theories of various types of algebraic structures as a whole. For example, universal algebra studies the overall theory of groups, as distinguished from studying particular groups.

History

As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many -- perhaps most -- of these problems were in some way related to the theory of algebraic equations. Major themes include:

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This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Abstract_algebra

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  We introduce quotient groups/factor groups. The quotient group of a group G by a normal subgroup H is the set of all cosets of H in G with the operation of coset multiplication. We prove the quotient group construction is indeed a group, and that it is a homomorphic image of the containing group. Thus, quotient groups give us a way to construct homomorphic images of a group. #abstractalgebra #grouptheory Normal Subgroups: https://youtu.be/kbT5SyF3H60 Left and Right Cosets in Normal Subgroups: https://youtu.be/XOu1zFutrMY Coset Multiplication on Normal Subgroups: https://youtu.be/DJyOdMBUdnM Fundamental Homomorphism Theorem: https://youtu.be/wtK1h9TIDCY Abstract Algebra Course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra Exercises: https://ww...

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11:24

Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra

• Order: Reorder
• Duration: 11:24
• Uploaded Date: 15 Dec 2018
• views: 396537

Normal subgroups are a powerful tool for creating factor groups (also called quotient groups). In this video we introduce the concept of a coset, talk about wh...

Normal subgroups are a powerful tool for creating factor groups (also called quotient groups). In this video we introduce the concept of a coset, talk about which subgroups are “normal” subgroups, and show when the collection of cosets can be treated as a group of their own. As a motivation, we will begin by discussing congruences. Our Abstract Algebra playlist is here: http://bit.ly/AbstractAlgebraSocratica Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ Ways to support our channel: ► Join our Patreon : https://www.patreon.com/socratica ► Make a one-time PayPal donation: https://www.paypal.me/socratica ► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9 Thank you! ♦♦♦♦♦♦♦♦♦♦ We recommend the following textbooks: Dummit & Foote, Abstract Algebra 3rd Edition http://amzn.to/2oOBd5S Milne, Algebra Course Notes (available free online) http://www.jmilne.org/math/CourseNotes/index.html ♦♦♦♦♦♦♦♦♦♦ Connect with us! Facebook: https://www.facebook.com/SocraticaStudios/ Instagram: https://www.instagram.com/SocraticaStudios/ Twitter: https://twitter.com/Socratica ♦♦♦♦♦♦♦♦♦♦ Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison ♦♦♦♦♦♦♦♦♦♦

https://wn.com/Normal_Subgroups_And_Quotient_Groups_(Aka_Factor_Groups)_Abstract_Algebra

Normal subgroups are a powerful tool for creating factor groups (also called quotient groups). In this video we introduce the concept of a coset, talk about which subgroups are “normal” subgroups, and show when the collection of cosets can be treated as a group of their own. As a motivation, we will begin by discussing congruences. Our Abstract Algebra playlist is here: http://bit.ly/AbstractAlgebraSocratica Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ Ways to support our channel: ► Join our Patreon : https://www.patreon.com/socratica ► Make a one-time PayPal donation: https://www.paypal.me/socratica ► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9 Thank you! ♦♦♦♦♦♦♦♦♦♦ We recommend the following textbooks: Dummit & Foote, Abstract Algebra 3rd Edition http://amzn.to/2oOBd5S Milne, Algebra Course Notes (available free online) http://www.jmilne.org/math/CourseNotes/index.html ♦♦♦♦♦♦♦♦♦♦ Connect with us! Facebook: https://www.facebook.com/SocraticaStudios/ Instagram: https://www.instagram.com/SocraticaStudios/ Twitter: https://twitter.com/Socratica ♦♦♦♦♦♦♦♦♦♦ Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison ♦♦♦♦♦♦♦♦♦♦

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Quotient Groups and Homomorphic Images | Abstract Algebra

• Order: Reorder
• Duration: 11:33
• Uploaded Date: 23 Jun 2023
• views: 8671

We introduce quotient groups/factor groups. The quotient group of a group G by a normal subgroup H is the set of all cosets of H in G with the operation of cose...

We introduce quotient groups/factor groups. The quotient group of a group G by a normal subgroup H is the set of all cosets of H in G with the operation of coset multiplication. We prove the quotient group construction is indeed a group, and that it is a homomorphic image of the containing group. Thus, quotient groups give us a way to construct homomorphic images of a group. #abstractalgebra #grouptheory Normal Subgroups: https://youtu.be/kbT5SyF3H60 Left and Right Cosets in Normal Subgroups: https://youtu.be/XOu1zFutrMY Coset Multiplication on Normal Subgroups: https://youtu.be/DJyOdMBUdnM Fundamental Homomorphism Theorem: https://youtu.be/wtK1h9TIDCY Abstract Algebra Course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Abstract Algebra: https://amzn.to/3IjoZaO Linear Algebra: https://amzn.to/43xAWEz Calculus: https://amzn.to/3PieD1M Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Loke Tan, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Roslyn Goddard, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon! Outro music is mine. You cannot find it anywhere, for now. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured

https://wn.com/Quotient_Groups_And_Homomorphic_Images_|_Abstract_Algebra

We introduce quotient groups/factor groups. The quotient group of a group G by a normal subgroup H is the set of all cosets of H in G with the operation of coset multiplication. We prove the quotient group construction is indeed a group, and that it is a homomorphic image of the containing group. Thus, quotient groups give us a way to construct homomorphic images of a group. #abstractalgebra #grouptheory Normal Subgroups: https://youtu.be/kbT5SyF3H60 Left and Right Cosets in Normal Subgroups: https://youtu.be/XOu1zFutrMY Coset Multiplication on Normal Subgroups: https://youtu.be/DJyOdMBUdnM Fundamental Homomorphism Theorem: https://youtu.be/wtK1h9TIDCY Abstract Algebra Course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Abstract Algebra: https://amzn.to/3IjoZaO Linear Algebra: https://amzn.to/43xAWEz Calculus: https://amzn.to/3PieD1M Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Loke Tan, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Roslyn Goddard, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon! Outro music is mine. You cannot find it anywhere, for now. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured

• published: 23 Jun 2023
• views: 8671

12:37

Chapter 5: Quotient groups | Essence of Group Theory

• Order: Reorder
• Duration: 12:37
• Uploaded Date: 01 Apr 2020
• views: 39564

Quotient groups is a very important concept in group theory, because it has paramount importance in group homomorphisms (connection with the isomorphism theorem...

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 fact, the concept of quotient groups is one way to define modular arithmetic formally, which allows us to prove a lot of number theory theorems once we draw parallels between group theory and number theory. For example, Fermat's little theorem and Euler's totient theorem are just corollaries of the Lagrange's theorem introduced in Chapter 3 of the video series. 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! 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! SUBSCRIBE and see you in the next video! #mathemanic #grouptheory #abstractalgebra #quotientgroup #intuition Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt For my contact email, check my About page on a PC. See you next time!

https://wn.com/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 fact, the concept of quotient groups is one way to define modular arithmetic formally, which allows us to prove a lot of number theory theorems once we draw parallels between group theory and number theory. For example, Fermat's little theorem and Euler's totient theorem are just corollaries of the Lagrange's theorem introduced in Chapter 3 of the video series. 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! 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! SUBSCRIBE and see you in the next video! #mathemanic #grouptheory #abstractalgebra #quotientgroup #intuition Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt For my contact email, check my About page on a PC. See you next time!

• published: 01 Apr 2020
• views: 39564

10:51

Abstract Algebra | Quotient Groups

• Order: Reorder
• Duration: 10:51
• Uploaded Date: 02 Mar 2020
• views: 24144

We introduce the notion of a quotient group and give some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

We introduce the notion of a quotient group and give some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

https://wn.com/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/

• published: 02 Mar 2020
• views: 24144

15:50

Why Normal Subgroups are Necessary for Quotient Groups

• Order: Reorder
• Duration: 15:50
• Uploaded Date: 27 Dec 2020
• views: 13493

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 nor...

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 why conjugation is so important to quotient groups! Group Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJHDvvls4OtoBHi6cNnTZ6a6 0:00 Intro to quotient groups 5:10 Not all subgroups work! 8:28 Quotient group condition 12:15 Normal subgroup examples Subscribe to see more new math videos! Music: OcularNebula - The Lopez

https://wn.com/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 why conjugation is so important to quotient groups! Group Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJHDvvls4OtoBHi6cNnTZ6a6 0:00 Intro to quotient groups 5:10 Not all subgroups work! 8:28 Quotient group condition 12:15 Normal subgroup examples Subscribe to see more new math videos! Music: OcularNebula - The Lopez

• published: 27 Dec 2020
• views: 13493

6:53

Quotient group example

• Order: Reorder
• Duration: 6:53
• Uploaded Date: 20 Feb 2018
• views: 5816

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.

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.

https://wn.com/Quotient_Group_Example

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.

• published: 20 Feb 2018
• views: 5816

13:11

Understanding and Calculating Factor (Quotient) Groups

• Order: Reorder
• Duration: 13:11
• Uploaded Date: 23 May 2014
• views: 10996

Busch

Busch

https://wn.com/Understanding_And_Calculating_Factor_(Quotient)_Groups

Busch

• published: 23 May 2014
• views: 10996

11:37

302.4A: Quotient Groups

• Order: Reorder
• Duration: 11:37
• Uploaded Date: 24 Jan 2013
• views: 29340

Reviewing the quotient of a group by a normal subgroup. The First and Fourth Isomorphism Theorems.

Reviewing the quotient of a group by a normal subgroup. The First and Fourth Isomorphism Theorems.

https://wn.com/302.4A_Quotient_Groups

Reviewing the quotient of a group by a normal subgroup. The First and Fourth Isomorphism Theorems.

• published: 24 Jan 2013
• views: 29340

42:55

Quotient Group & Homomorphism II | Group Theory - Concepts & Questions | WB-SET | TS-SET | K-SET

• Order: Reorder
• Duration: 42:55
• Uploaded Date: 17 Aug 2024
• views: 49

IFAS: India's No. 1 Institute for the SET, CSIR NET & GATE Examination!! Dear Aspirants, To get detailed info about SET Courses from our Academic Experts, ple...

IFAS: India's No. 1 Institute for the SET, CSIR NET & GATE Examination!! Dear Aspirants, To get detailed info about SET Courses from our Academic Experts, please fill the form: https://bit.ly/setlifescience Also, you can call on 9172266888 for any support. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Download IFAS App for Free Lectures, PYQ & Practice Questions: 👇 📲Android: https://bit.ly/3TGYe6l ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ You can enroll in IFAS Subscription and get the following benefits: 👇 ✅Live Interactive lectures with India’s Best Educators for the SET & CSIR NET Exam. ✅You can ask doubts in live online classes. ✔️Clear your doubts after class as well with Expert Faculty Members using Chat & Audio Functionality ✅Recorded Video lectures can be downloaded in IFAS App (Android, iOS) & watch offline. ✅Online Test Series (Unit Test & Mock Test) with detailed analysis. ✅Teacher's eNote will be available in IFAS App. ✅ Daily Practice Sheet for each lecture to crystal clear concepts. ✅Previous Year Question with detailed explanation. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/ifas_set_prep ✔Instagram: https://www.instagram.com/ifas_set_exam_prep/ ✔Facebook Page: https://www.facebook.com/profile.php?id=100088056864827 ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ set exam set exam 2024 mh set exam paper 1 mh set exam 2024 NE-SLET Exam ne slet 2024 set exam update 2024 set exam study tips set exam preparation set 2024 examination mh set 2024 exam date maharashtra set exam 2024 mh set exam 2024 notification mh set exam application form 2024 set exam details set exam general aptitude set exam preparation for paper 1 set exam paper 1 preparation set general aptitude preparation strategy ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ #setexam #setaptitude #setexampreparation #setpaper1 #ifasmhset #ifas

https://wn.com/Quotient_Group_Homomorphism_Ii_|_Group_Theory_Concepts_Questions_|_Wb_Set_|_Ts_Set_|_K_Set

IFAS: India's No. 1 Institute for the SET, CSIR NET & GATE Examination!! Dear Aspirants, To get detailed info about SET Courses from our Academic Experts, please fill the form: https://bit.ly/setlifescience Also, you can call on 9172266888 for any support. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Download IFAS App for Free Lectures, PYQ & Practice Questions: 👇 📲Android: https://bit.ly/3TGYe6l ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ You can enroll in IFAS Subscription and get the following benefits: 👇 ✅Live Interactive lectures with India’s Best Educators for the SET & CSIR NET Exam. ✅You can ask doubts in live online classes. ✔️Clear your doubts after class as well with Expert Faculty Members using Chat & Audio Functionality ✅Recorded Video lectures can be downloaded in IFAS App (Android, iOS) & watch offline. ✅Online Test Series (Unit Test & Mock Test) with detailed analysis. ✅Teacher's eNote will be available in IFAS App. ✅ Daily Practice Sheet for each lecture to crystal clear concepts. ✅Previous Year Question with detailed explanation. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/ifas_set_prep ✔Instagram: https://www.instagram.com/ifas_set_exam_prep/ ✔Facebook Page: https://www.facebook.com/profile.php?id=100088056864827 ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ set exam set exam 2024 mh set exam paper 1 mh set exam 2024 NE-SLET Exam ne slet 2024 set exam update 2024 set exam study tips set exam preparation set 2024 examination mh set 2024 exam date maharashtra set exam 2024 mh set exam 2024 notification mh set exam application form 2024 set exam details set exam general aptitude set exam preparation for paper 1 set exam paper 1 preparation set general aptitude preparation strategy ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ #setexam #setaptitude #setexampreparation #setpaper1 #ifasmhset #ifas

• published: 17 Aug 2024
• views: 49

11:08

Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra

• Order: Reorder
• Duration: 11:08
• Uploaded Date: 04 Mar 2020
• views: 289251

📒⏩Comment Below If This Video Helped You 💯 Like 👍 & Share With Your Classmates - ALL THE BEST 🔥 Do Visit My Second Channel - https://bit.ly/3rMGcSA This video...

📒⏩Comment Below If This Video Helped You 💯 Like 👍 & Share With Your Classmates - ALL THE BEST 🔥 Do Visit My Second Channel - https://bit.ly/3rMGcSA This video lecture of Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra | Examples & Solution By Definition | Problems & Concepts by GP Sir will help Engineering and Basic Science students to understand following topic of Mathematics: 1. What is Quotient Group ? 2. How To Find Quotient Group G/H ? 3. Examples Of Quotient Group. 4. This is Part Of Abstract Algebra #GroupTheory #QuotientGroup #Examples #AbstractAlgebra #EngineeringMahemaics #BSCMaths #GATE #IITJAM #CSIRNET This Concept is very important in Engineering & Basic Science Students. This video is very useful for B.Sc./B.Tech & M.Sc./M.Tech. students also preparing NET, GATE and IIT-JAM Aspirants. Find Online Solutions Of Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra | Problems & Concepts by GP Sir (Gajendra Purohit) Do Like & Share this Video with your Friends. If you are watching for the first time then Subscribe to our Channel and stay updated for more videos around Mathematics Time Stamp Quotient group : Theorems & Examples- 0:00 | An intro 0:18 | Topic introduction 1:19 | Quotient group definition 4:03 | Example 1 6:35 | Example 2 9:58 | Conclusion of video 📚 Buy My Book For CSIR NET Mathematics: https://amzn.to/30H9HcD (Best Seller) 🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹 💯 My Social Media Handles 💯 📌 GP Sir Instagram: https://www.instagram.com/dr.gajendrapurohit 📌 GP Sir Facebook Page: https://www.facebook.com/drgpsir 📌 📚 Important Course Playlist 📚 📌 Link to B.Sc. Maths Playlist: https://bit.ly/3cAg1YI 📌 Link to Engineering Maths Playlist: https://bit.ly/3thNYUK 📌 Link to IIT-JAM Maths Playlist: https://bit.ly/3tiBpZl 📌 Link to GATE (Engg.) Maths Playlist: https://bit.ly/3eEI3VC 📌 Link to IAS Optional Maths Playlist: https://bit.ly/3vzHl2a 📌 Link To CSIR NET Maths Playlist: https://bit.ly/3rMHe0U 🔥 Motivational Videos & Tips For Students (Make Student Life Better) - https://bit.ly/3tdAGbM 📷 My Equipment & Gear 📷 ▶️My Phone - https://amzn.to/38Cfvsg ▶️My Primary Laptop - https://amzn.to/2PUW2MG ▶️My Secondary Laptop - https://amzn.to/38EHQy0 ▶️My Primary Camera - https://amzn.to/3eFl9NN ▶️My Secondary Camera - https://amzn.to/3vmBs8h ▶️Secondary Mic - https://amzn.to/2PSVffd ▶️Vlogging Mic - https://amzn.to/38EIz2g ▶️Tripod - https://amzn.to/3ctwJJn ▶️Secondary Screen - https://amzn.to/38FCYZw 📚 Following Topics Are Also Available 📚 📒Linear Algebra: https://bit.ly/3qMKgB0 📕Abstract Algebra Lectures: https://bit.ly/3rOh0uS 📗Real Analysis: https://bit.ly/3tetewY 📘Complex Analysis: https://bit.ly/3vnBk8D 📙Differential Equation: https://bit.ly/38FnAMH 📒Partial Differentiation: https://bit.ly/3tkNaOV 📕Numerical Analysis: https://bit.ly/3vrlEkA 📗Operation Research: https://bit.ly/3cvBxOq 📘Statistics & Probability: https://bit.ly/3qMf3hf 📙Integral Calculus: https://bit.ly/3qIOtFz 📒Differential Calculus: https://bit.ly/3bM9CKT 📕Multivariable Calculus: https://bit.ly/3qOsEEA 📗Vector Calculus: https://bit.ly/2OvpEjv Thanks For Watching My Video 🙏🏻 Like, Share & Subscribe 💯 Dr.Gajendra Purohit

https://wn.com/Group_Theory_|_Quotient_Group_|_Quotient_Group_Examples_|_Abstract_Algebra

📒⏩Comment Below If This Video Helped You 💯 Like 👍 & Share With Your Classmates - ALL THE BEST 🔥 Do Visit My Second Channel - https://bit.ly/3rMGcSA This video lecture of Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra | Examples & Solution By Definition | Problems & Concepts by GP Sir will help Engineering and Basic Science students to understand following topic of Mathematics: 1. What is Quotient Group ? 2. How To Find Quotient Group G/H ? 3. Examples Of Quotient Group. 4. This is Part Of Abstract Algebra #GroupTheory #QuotientGroup #Examples #AbstractAlgebra #EngineeringMahemaics #BSCMaths #GATE #IITJAM #CSIRNET This Concept is very important in Engineering & Basic Science Students. This video is very useful for B.Sc./B.Tech & M.Sc./M.Tech. students also preparing NET, GATE and IIT-JAM Aspirants. Find Online Solutions Of Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra | Problems & Concepts by GP Sir (Gajendra Purohit) Do Like & Share this Video with your Friends. If you are watching for the first time then Subscribe to our Channel and stay updated for more videos around Mathematics Time Stamp Quotient group : Theorems & Examples- 0:00 | An intro 0:18 | Topic introduction 1:19 | Quotient group definition 4:03 | Example 1 6:35 | Example 2 9:58 | Conclusion of video 📚 Buy My Book For CSIR NET Mathematics: https://amzn.to/30H9HcD (Best Seller) 🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹 💯 My Social Media Handles 💯 📌 GP Sir Instagram: https://www.instagram.com/dr.gajendrapurohit 📌 GP Sir Facebook Page: https://www.facebook.com/drgpsir 📌 📚 Important Course Playlist 📚 📌 Link to B.Sc. Maths Playlist: https://bit.ly/3cAg1YI 📌 Link to Engineering Maths Playlist: https://bit.ly/3thNYUK 📌 Link to IIT-JAM Maths Playlist: https://bit.ly/3tiBpZl 📌 Link to GATE (Engg.) Maths Playlist: https://bit.ly/3eEI3VC 📌 Link to IAS Optional Maths Playlist: https://bit.ly/3vzHl2a 📌 Link To CSIR NET Maths Playlist: https://bit.ly/3rMHe0U 🔥 Motivational Videos & Tips For Students (Make Student Life Better) - https://bit.ly/3tdAGbM 📷 My Equipment & Gear 📷 ▶️My Phone - https://amzn.to/38Cfvsg ▶️My Primary Laptop - https://amzn.to/2PUW2MG ▶️My Secondary Laptop - https://amzn.to/38EHQy0 ▶️My Primary Camera - https://amzn.to/3eFl9NN ▶️My Secondary Camera - https://amzn.to/3vmBs8h ▶️Secondary Mic - https://amzn.to/2PSVffd ▶️Vlogging Mic - https://amzn.to/38EIz2g ▶️Tripod - https://amzn.to/3ctwJJn ▶️Secondary Screen - https://amzn.to/38FCYZw 📚 Following Topics Are Also Available 📚 📒Linear Algebra: https://bit.ly/3qMKgB0 📕Abstract Algebra Lectures: https://bit.ly/3rOh0uS 📗Real Analysis: https://bit.ly/3tetewY 📘Complex Analysis: https://bit.ly/3vnBk8D 📙Differential Equation: https://bit.ly/38FnAMH 📒Partial Differentiation: https://bit.ly/3tkNaOV 📕Numerical Analysis: https://bit.ly/3vrlEkA 📗Operation Research: https://bit.ly/3cvBxOq 📘Statistics & Probability: https://bit.ly/3qMf3hf 📙Integral Calculus: https://bit.ly/3qIOtFz 📒Differential Calculus: https://bit.ly/3bM9CKT 📕Multivariable Calculus: https://bit.ly/3qOsEEA 📗Vector Calculus: https://bit.ly/2OvpEjv Thanks For Watching My Video 🙏🏻 Like, Share & Subscribe 💯 Dr.Gajendra Purohit

• published: 04 Mar 2020
• views: 289251

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11:24

Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra

Normal subgroups are a powerful tool for creating factor groups (also called quotient grou...

published: 15 Dec 2018

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Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra

Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra

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• published: 15 Dec 2018
• views: 396537

Normal subgroups are a powerful tool for creating factor groups (also called quotient groups). In this video we introduce the concept of a coset, talk about which subgroups are “normal” subgroups, and show when the collection of cosets can be treated as a group of their own. As a motivation, we will begin by discussing congruences. Our Abstract Algebra playlist is here: http://bit.ly/AbstractAlgebraSocratica Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ Ways to support our channel: ► Join our Patreon : https://www.patreon.com/socratica ► Make a one-time PayPal donation: https://www.paypal.me/socratica ► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9 Thank you! ♦♦♦♦♦♦♦♦♦♦ We recommend the following textbooks: Dummit & Foote, Abstract Algebra 3rd Edition http://amzn.to/2oOBd5S Milne, Algebra Course Notes (available free online) http://www.jmilne.org/math/CourseNotes/index.html ♦♦♦♦♦♦♦♦♦♦ Connect with us! Facebook: https://www.facebook.com/SocraticaStudios/ Instagram: https://www.instagram.com/SocraticaStudios/ Twitter: https://twitter.com/Socratica ♦♦♦♦♦♦♦♦♦♦ Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison ♦♦♦♦♦♦♦♦♦♦

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11:33

Quotient Groups and Homomorphic Images | Abstract Algebra

We introduce quotient groups/factor groups. The quotient group of a group G by a normal su...

published: 23 Jun 2023

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Quotient Groups and Homomorphic Images | Abstract Algebra

Quotient Groups and Homomorphic Images | Abstract Algebra

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• published: 23 Jun 2023
• views: 8671

We introduce quotient groups/factor groups. The quotient group of a group G by a normal subgroup H is the set of all cosets of H in G with the operation of coset multiplication. We prove the quotient group construction is indeed a group, and that it is a homomorphic image of the containing group. Thus, quotient groups give us a way to construct homomorphic images of a group. #abstractalgebra #grouptheory Normal Subgroups: https://youtu.be/kbT5SyF3H60 Left and Right Cosets in Normal Subgroups: https://youtu.be/XOu1zFutrMY Coset Multiplication on Normal Subgroups: https://youtu.be/DJyOdMBUdnM Fundamental Homomorphism Theorem: https://youtu.be/wtK1h9TIDCY Abstract Algebra Course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Abstract Algebra: https://amzn.to/3IjoZaO Linear Algebra: https://amzn.to/43xAWEz Calculus: https://amzn.to/3PieD1M Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Loke Tan, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Roslyn Goddard, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon! Outro music is mine. You cannot find it anywhere, for now. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured

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12:37

Chapter 5: Quotient groups | Essence of Group Theory

Quotient groups is a very important concept in group theory, because it has paramount impo...

published: 01 Apr 2020

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Chapter 5: Quotient groups | Essence of Group Theory

Chapter 5: Quotient groups | Essence of Group Theory

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• published: 01 Apr 2020
• views: 39564

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 fact, the concept of quotient groups is one way to define modular arithmetic formally, which allows us to prove a lot of number theory theorems once we draw parallels between group theory and number theory. For example, Fermat's little theorem and Euler's totient theorem are just corollaries of the Lagrange's theorem introduced in Chapter 3 of the video series. 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! 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! SUBSCRIBE and see you in the next video! #mathemanic #grouptheory #abstractalgebra #quotientgroup #intuition Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt For my contact email, check my About page on a PC. See you next time!

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10:51

Abstract Algebra | Quotient Groups

We introduce the notion of a quotient group and give some examples. http://www.michael-pe...

published: 02 Mar 2020

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Abstract Algebra | Quotient Groups

Abstract Algebra | Quotient Groups

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• published: 02 Mar 2020
• views: 24144

We introduce the notion of a quotient group and give some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

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15:50

Why Normal Subgroups are Necessary for Quotient Groups

Proof that cosets are disjoint: https://youtu.be/uxhAUmgSHnI In order for a subgroup to ...

published: 27 Dec 2020

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Why Normal Subgroups are Necessary for Quotient Groups

Why Normal Subgroups are Necessary for Quotient Groups

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• published: 27 Dec 2020
• views: 13493

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 why conjugation is so important to quotient groups! Group Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJHDvvls4OtoBHi6cNnTZ6a6 0:00 Intro to quotient groups 5:10 Not all subgroups work! 8:28 Quotient group condition 12:15 Normal subgroup examples Subscribe to see more new math videos! Music: OcularNebula - The Lopez

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6:53

Quotient group example

Now that we know what a quotient group is, let's take a look at an example to cement our u...

published: 20 Feb 2018

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Quotient group example

Quotient group example

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• published: 20 Feb 2018
• views: 5816

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.

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13:11

Understanding and Calculating Factor (Quotient) Groups

Busch

published: 23 May 2014

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Understanding and Calculating Factor (Quotient) Groups

Understanding and Calculating Factor (Quotient) Groups

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• published: 23 May 2014
• views: 10996

Busch

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11:37

302.4A: Quotient Groups

Reviewing the quotient of a group by a normal subgroup. The First and Fourth Isomorphism T...

published: 24 Jan 2013

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302.4A: Quotient Groups

302.4A: Quotient Groups

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• published: 24 Jan 2013
• views: 29340

Reviewing the quotient of a group by a normal subgroup. The First and Fourth Isomorphism Theorems.

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42:55

Quotient Group & Homomorphism II | Group Theory - Concepts & Questions | WB-SET | TS-SET | K-SET

IFAS: India's No. 1 Institute for the SET, CSIR NET & GATE Examination!! Dear Aspirants, ...

published: 17 Aug 2024

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Quotient Group & Homomorphism II | Group Theory - Concepts & Questions | WB-SET | TS-SET | K-SET

Quotient Group & Homomorphism II | Group Theory - Concepts & Questions | WB-SET | TS-SET | K-SET

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• published: 17 Aug 2024
• views: 49

IFAS: India's No. 1 Institute for the SET, CSIR NET & GATE Examination!! Dear Aspirants, To get detailed info about SET Courses from our Academic Experts, please fill the form: https://bit.ly/setlifescience Also, you can call on 9172266888 for any support. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Download IFAS App for Free Lectures, PYQ & Practice Questions: 👇 📲Android: https://bit.ly/3TGYe6l ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ You can enroll in IFAS Subscription and get the following benefits: 👇 ✅Live Interactive lectures with India’s Best Educators for the SET & CSIR NET Exam. ✅You can ask doubts in live online classes. ✔️Clear your doubts after class as well with Expert Faculty Members using Chat & Audio Functionality ✅Recorded Video lectures can be downloaded in IFAS App (Android, iOS) & watch offline. ✅Online Test Series (Unit Test & Mock Test) with detailed analysis. ✅Teacher's eNote will be available in IFAS App. ✅ Daily Practice Sheet for each lecture to crystal clear concepts. ✅Previous Year Question with detailed explanation. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/ifas_set_prep ✔Instagram: https://www.instagram.com/ifas_set_exam_prep/ ✔Facebook Page: https://www.facebook.com/profile.php?id=100088056864827 ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ set exam set exam 2024 mh set exam paper 1 mh set exam 2024 NE-SLET Exam ne slet 2024 set exam update 2024 set exam study tips set exam preparation set 2024 examination mh set 2024 exam date maharashtra set exam 2024 mh set exam 2024 notification mh set exam application form 2024 set exam details set exam general aptitude set exam preparation for paper 1 set exam paper 1 preparation set general aptitude preparation strategy ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ #setexam #setaptitude #setexampreparation #setpaper1 #ifasmhset #ifas

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11:08

Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra

📒⏩Comment Below If This Video Helped You 💯 Like 👍 & Share With Your Classmates - ALL THE ...

published: 04 Mar 2020

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Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra

Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra

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• published: 04 Mar 2020
• views: 289251

📒⏩Comment Below If This Video Helped You 💯 Like 👍 & Share With Your Classmates - ALL THE BEST 🔥 Do Visit My Second Channel - https://bit.ly/3rMGcSA This video lecture of Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra | Examples & Solution By Definition | Problems & Concepts by GP Sir will help Engineering and Basic Science students to understand following topic of Mathematics: 1. What is Quotient Group ? 2. How To Find Quotient Group G/H ? 3. Examples Of Quotient Group. 4. This is Part Of Abstract Algebra #GroupTheory #QuotientGroup #Examples #AbstractAlgebra #EngineeringMahemaics #BSCMaths #GATE #IITJAM #CSIRNET This Concept is very important in Engineering & Basic Science Students. This video is very useful for B.Sc./B.Tech & M.Sc./M.Tech. students also preparing NET, GATE and IIT-JAM Aspirants. Find Online Solutions Of Group Theory | Quotient Group | Quotient Group Examples | Abstract Algebra | Problems & Concepts by GP Sir (Gajendra Purohit) Do Like & Share this Video with your Friends. If you are watching for the first time then Subscribe to our Channel and stay updated for more videos around Mathematics Time Stamp Quotient group : Theorems & Examples- 0:00 | An intro 0:18 | Topic introduction 1:19 | Quotient group definition 4:03 | Example 1 6:35 | Example 2 9:58 | Conclusion of video 📚 Buy My Book For CSIR NET Mathematics: https://amzn.to/30H9HcD (Best Seller) 🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹🔸🔹 💯 My Social Media Handles 💯 📌 GP Sir Instagram: https://www.instagram.com/dr.gajendrapurohit 📌 GP Sir Facebook Page: https://www.facebook.com/drgpsir 📌 📚 Important Course Playlist 📚 📌 Link to B.Sc. Maths Playlist: https://bit.ly/3cAg1YI 📌 Link to Engineering Maths Playlist: https://bit.ly/3thNYUK 📌 Link to IIT-JAM Maths Playlist: https://bit.ly/3tiBpZl 📌 Link to GATE (Engg.) Maths Playlist: https://bit.ly/3eEI3VC 📌 Link to IAS Optional Maths Playlist: https://bit.ly/3vzHl2a 📌 Link To CSIR NET Maths Playlist: https://bit.ly/3rMHe0U 🔥 Motivational Videos & Tips For Students (Make Student Life Better) - https://bit.ly/3tdAGbM 📷 My Equipment & Gear 📷 ▶️My Phone - https://amzn.to/38Cfvsg ▶️My Primary Laptop - https://amzn.to/2PUW2MG ▶️My Secondary Laptop - https://amzn.to/38EHQy0 ▶️My Primary Camera - https://amzn.to/3eFl9NN ▶️My Secondary Camera - https://amzn.to/3vmBs8h ▶️Secondary Mic - https://amzn.to/2PSVffd ▶️Vlogging Mic - https://amzn.to/38EIz2g ▶️Tripod - https://amzn.to/3ctwJJn ▶️Secondary Screen - https://amzn.to/38FCYZw 📚 Following Topics Are Also Available 📚 📒Linear Algebra: https://bit.ly/3qMKgB0 📕Abstract Algebra Lectures: https://bit.ly/3rOh0uS 📗Real Analysis: https://bit.ly/3tetewY 📘Complex Analysis: https://bit.ly/3vnBk8D 📙Differential Equation: https://bit.ly/38FnAMH 📒Partial Differentiation: https://bit.ly/3tkNaOV 📕Numerical Analysis: https://bit.ly/3vrlEkA 📗Operation Research: https://bit.ly/3cvBxOq 📘Statistics & Probability: https://bit.ly/3qMf3hf 📙Integral Calculus: https://bit.ly/3qIOtFz 📒Differential Calculus: https://bit.ly/3bM9CKT 📕Multivariable Calculus: https://bit.ly/3qOsEEA 📗Vector Calculus: https://bit.ly/2OvpEjv Thanks For Watching My Video 🙏🏻 Like, Share & Subscribe 💯 Dr.Gajendra Purohit

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Quotient group

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 the group structure. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n 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.

In a quotient of 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 G / N, where G is the original group and N is the normal subgroup. (This is pronounced "G mod N," where "mod" 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 G. Specifically, the image of G under a homomorphism φ: G → H is isomorphic to G / ker(φ) where ker(φ) denotes the kernel of φ.

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This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Quotient_group

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