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Direct Products of Groups  (Abstract Algebra)
 
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The direct product is a way to combine two groups into a new, larger group. Just as you can factor integers into prime numbers, you can break apart some groups into a direct product of simpler groups. If​ ​you​’d​ ​like​ ​to​ ​help​ ​us​ ​make​ ​videos more quickly,​ ​you​ ​can​ ​support​ ​us​ on ​Patreon​ at https://www.patreon.com/socratica We​ ​also​ ​welcome​ ​Bitcoin​ ​donations!​ ​​ ​Our​ ​Bitcoin​ ​address​ ​is: 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 ************** Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W You​ ​can​ ​also​ ​follow​ ​Socratica​ ​on: -​ ​Twitter:​ ​@socratica -​ ​Instagram:​ ​@SocraticaStudios -​ ​Facebook:​ ​@SocraticaStudios ******** Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison
Views: 56101 Socratica
External Direct Product part 1
 
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Definition and theorems related to External Direct Product valuable for exam as well
Views: 2391 Reesha Education
Group Theory 35, Internal Direct Products
 
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Group Theory 35, Internal Direct Products http://groupprops.subwiki.org/wiki/Internal_direct_product
Views: 3799 LadislauFernandes
Lec - 40 Cartesian Product of Groups or Direct Product of Groups | IIT JAM | CSIR UGC NET | GATE MA
 
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This lecture contains External Direct Product and Internal Direct Product of Groups(Here Subgroup must be Normal and HK=G and H intersection K={e}). It also contains properties of the Cartesian product of groups and its applications. For Internal Direct Product please go through the lecture link given below https://www.youtube.com/watch?v=JR3b9FsZUx8&list=PL8WTUM4A_UNkZmDtNIxA_xU_lj4BqVtSn&index=56 #cartesianproductofgroups #directproductofgroups #grouptheory #abstractalgebra #csirnetmaths #iitjammsths
Views: 1340 Mathematical Science
Direct product | Group theory | Reducible representation
 
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In this lecture we will learn to calculate the combination of irreducible representation in direct product question from group theory chemistry. Important for csir net chemical science and Gate chemistry. Lecture by Csir net gate chemistry.
Internal and External Direct Products Part 1
 
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In this video we discuss the definition of the internal and external direct products.
Views: 7917 Ben1994
Chapter8: External direct products -- Discussion Chapter9: Normal Subgroups and Factor groups - 1
 
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اسم المساق: جبر حديث 1 اسم المحاضر: د.إسماعيل محيي الدين الأسطل، و من المحاضرة 17 إلى المحاضرة 30 أ. محمود السرساوي ، إكمال لمحاضرات الدكتور إسماعيل الأسطل خلال رحلته للحج والعمرة مشرف الموقع: أ. خالد محمد الأدغم الكلية: العلوم القسم: الرياضيات وصف المساق : يدرس المساق العمليات الثنائية، الزمر، الزمر الدورية، الزمر الجزئية، الضرب المباشر، التشاكل، الزمر المتماثلة، علاقات التكافؤ، المرفقات، عد عناصر الزمرة المنهية ونظرية لاجرانج، الزمر الجزئية القياسية، النظرية الأساسية للزمر التبديلية. مركز التميز الأكاديمي: http://lectures.iugaza.edu.ps/ http://quality.iugaza.edu.ps/ExcellenceCenter/ar/Default.aspx قائمة محاضرات المساق: http://www.youtube.com/playlist?list=PL9fwy3NUQKwZKOpj354PRgwYPWWgxchnI الجامعة الإسلامية غزة: http://www.iugaza.edu.ps/ar قناة مركز التميز الأكاديمي بالجامعة الإسلامية - غزة على YouTube : http://www.youtube.com/iugaza1
Views: 4426 iugaza1
What is Direct product of groups?, Explain Direct product of groups, Define Direct product of groups
 
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#Directproductofgroups #audioversity ~~~ Direct product of groups ~~~ Title: What is Direct product of groups?, Explain Direct product of groups, Define Direct product of groups Created on: 2019-01-07 Source Link: https://en.wikipedia.org/wiki/Direct_product_of_groups ------ Description: In group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. ------ To see your favorite topic here, fill out this request form: https://docs.google.com/forms/d/e/1FAIpQLScU0dLbeWsc01IC0AaO8sgaSgxMFtvBL31c_pjnwEZUiq99Fw/viewform ------ Source: Wikipedia.org articles, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license. Support: Donations can be made from https://wikimediafoundation.org/wiki/Ways_to_Give to support Wikimedia Foundation and knowledge sharing.
Views: 2 Audioversity
Group Theory 33, Properties of External Direct products
 
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Group Theory 33, Properties of External Direct products
Views: 6104 LadislauFernandes
Direct Products of Groups
 
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We define the product of two groups and prove that the operation on the product does indeed define a group structure.
Views: 6048 Tim Hodges
direct product of groups part 1 english
 
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There are three parts of this lecture. In all the three lectures following things has been explained: (i) External Direct Product of two groups. (ii) Internal Direct Product of two groups. (iii) External direct product of n copies of groups. (iv) Internal direct product of n copies of groups. (v) Internal direct product of countable many copies of groups. (vi) difference between direct product and direct sum of groups.
Views: 183 Vivek Kumar Jain
301.9F Internal Direct Product of Groups
 
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Groups that "form their own times tables" can be defined as internal direct products of two of their subgroups. Here's how that works, and why the result is isomorphic to an external direct product as well.
Views: 100 Matthew Salomone
direct product of groups part 3
 
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This is the third and last lecture in the series of lectures Direct Product of Groups. In this lecture direct product of n copies of groups is explained and the difference between direct product and direct sum is explained.
Views: 261 Vivek Kumar Jain
Group Theory, 2nd Paper ,External Direct Product,
 
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Here I have discuss about External Direct Product(Definition with Example)
Views: 62 point Creation
Abstract Algebra, Lec 18A, External Direct Products Calculations & Isomorphisms, Normal Subgroups
 
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(0:00) Lecture plan. (0:28) Criterion for when the external direct product of cyclic groups is cyclic and Example (Z4 + Z9 is isomorphic to Z36). (6:40) Extension to the external direct product of more than two cyclic groups (Example: Z2 + Z3 + Z25 is isomorphic to Z150). (7:27) When the factors have orders that are not relatively prime, the resulting group will not be cyclic (Example: Z4 + Z6 is not cyclic). (8:40) Example: Find the number of elements of various orders in Z8 + Z8. (12:23) Isomorphism facts about U-groups (group of units modulo n); Examples: U(385) is isomorphic to Z4 + Z6 + Z10 and U(225) is isomorphic to Z6 + Z20. (19:39) Facts about subgroups of U-groups. (20:11) Definition of a normal subgroup of a group (in terms of left and right coset equality). (21:30) Basic facts and examples. (24:12) Subgroups of index 2 in a group are normal in that group. (26:28) Transitivity might not be satisfied. (27:40) Normal subgroup test. (29:20) For any subgroup H of G and any x in G, the set xHx^(-1) is a subgroup of G. (30:33) Proof of this last fact using the two-step subgroup test.
Views: 644 Bill Kinney
Subgroups and Direct Products of U(n)
 
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We define a subset of U(n) that turns out to be a subgroup. Then, we use that definition to look at some isomorphism results about U(n).
Views: 32 Patrick Jones
301.8I Extra: Semi-Direct Products of Groups
 
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In a semi-direct product of groups, elements from the second factor are permitted to "cross over" and interfere with the arithmetic in the first factor via an automorphism. Here's the definition and a brief example of a semi-direct product construction of Z4 with Z4.
Views: 150 Matthew Salomone
Direct Products of Finite Cyclic Groups Video 1
 
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Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Direct Products of Finite Cyclic Groups Video 1. How to determine if a direct product of finite cyclic groups is itself cyclic. This video has very easy examples.
Views: 3318 The Math Sorcerer
Direct product representation (CHE)
 
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Subject: Chemistry Paper: Applications of molecular symmetry and group theory Module: Direct product representation Content writer: Prof B S Garg
Views: 982 Vidya-mitra
Modern Algebra (Abstract Algebra) Made Easy-Part 7-Direct Products/Finitely Generated Abelian Groups
 
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http://www.pensieve.net/course/13 In this lecture, I define and explain in detail what Finitely Generated Abelian Groups are. I give examples, proofs, and some interesting tidbits that are hard to come by. Hope you like it! Works Cited: The only significant source for these videos is 'A First Course in Abstract Algebra', 2nd ed. by John Fraleigh. Though most of the problems and definitions come from this book, I explain the all the material & solutions in my own words.
Views: 29833 scalenescott
Lec - 41 Order of Elements in Cartesian Product of Groups | IIT JAM | CSIR UGC NET | GATE MA | B Sc
 
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Easy way to find the order of elements in direct product of groups #orderofelementsofcartesiangroup #orderofelemets #directproductofgroups #cartesianproductofgroups #grouptheory #abstractalgebra
Views: 1641 Mathematical Science
Abstract Alg, Lec 17B, External Direct Product Order Calculations, Isomorphisms, Normal Subgroups
 
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(0:00) Correct mistake from the end of Lecture 17A on the number of elements of order 4 in the external direct product of Z4 with itself. (0:24) Example: Number of elements of various orders in Z2 + Z2 + Z4 (external direct product). (4:43) Example: Number of elements of various orders in Z2 + Z2 + Z2 + Z2 (external direct product). (5:40) Example: Number of elements of various orders in Z2 + Z8 (external direct product). (8:05) All these examples are groups which are not isomorphic (and none are cyclic). (9:06) Criterion for the external direct product of two finite cyclic groups to be cyclic and corollaries. (11:28) The order of the groups in an external direct product is irrelevant (as far as isomorphism classes go). (12:59) Isomorphism facts related to U-Groups (groups of units modulo n) and examples: U(70),
Views: 383 Bill Kinney
Quantum Chemistry 12.15 - Irrep Direct Products
 
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Short lecture on irrep direct products. The irreducible representations of a point group form a group themeselves through their multiplication. This is relevant in molecular term symbols, where the direct product of all molecular orbital irrep labels results in the irrep of the total wavefunction, for example singlet A1 as the ground state of H2O. Symmetry at Otterbein: http://symmetry.otterbein.edu Character Tables: http://symmetry.jacobs-university.de/ Notes Slide: http://i.imgur.com/IxfRDlf.png --- About TMP Chem --- All TMP Chem content is free for everyone, everywhere, and created independently by Trent Parker. Email: [email protected] --- Video Links --- Chapter Playlist: https://www.youtube.com/playlist?list=PLm8ZSArAXicKX9g2IA7FAe0pihiwJCjLG Course Playlist: https://www.youtube.com/playlist?list=PLm8ZSArAXicL3jKr_0nHHs5TwfhdkMFhh Course Review: https://www.youtube.com/playlist?list=PLm8ZSArAXicLTRn3cJyyU1TiU7n_Pp4X1 Other Courses: https://www.youtube.com/playlist?list=PLm8ZSArAXicIXArfap9Tcb8izqRPvE0BK Channel Info: https://www.youtube.com/playlist?list=PLm8ZSArAXicLlGO4Rvpz-D6vX8MFbOn4V --- Social Links --- Facebook: https://www.facebook.com/tmpchem Twitter: https://www.twitter.com/tmpchem LinkedIn: https://www.linkedin.com/in/tmpchem Imgur: https://tmpchem.imgur.com GitHub: https://www.github.com/tmpchem --- Equipment --- Microphone: Blue Yeti USB Microphone Drawing Tablet: Wacom Intuos Pen and Touch Small Drawing Program: Autodesk Sketchbook Express Screen Capture: Corel Visual Studio Pro X8
Views: 4290 TMP Chem
Direct product of groups
 
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Direct product of groups In group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H.This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. =======Image-Copyright-Info======== License: Creative Commons Attribution-Share Alike 3.0 (CC BY-SA 3.0) LicenseLink: http://creativecommons.org/licenses/by-sa/3.0 Author-Info: derivative work: Pbroks13 (talk) Cyclic_group.png: Jakob.scholbach Image Source: https://en.wikipedia.org/wiki/File:Cyclic_group.svg =======Image-Copyright-Info======== -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=szHP0cnyq-o
Views: 390 WikiAudio
301.8B Defining, Motivating, and Modeling Direct Products
 
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The direct product of two groups G and H is a group in which the elements and algebra of both G and H coexist side-by-side in an ordered pair. We look at two models for the direct product of cyclic groups, trying to tell the difference between two examples of order 12.
Views: 137 Matthew Salomone
Group Theory 32, External Direct Products
 
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Group Theory 32, External Direct Products
Views: 7275 LadislauFernandes
first lecture internal direct product by sir kamil nawaz
 
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INTERNAL DIRECT PRODCUT FIRST LECTURE BY SIR KAMIL NAWAZ GROUP THEORY ADVANCE ALGEBRA MSC MATHEMATICS DEFINITION OF INTERNAL DIRECT PRODDUCT
Views: 2446 kamil nawaz
Lec - 15 Application of Sylow's Theorem || Internal Direct Product || Part V | IIT JAM | NET | GATE
 
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Application of Sylow's Theorem with Internal Direct Product Why group of order 15, 35, 65, 91 are cyclic? Why group of order 1001 is Abelian? #applicationofsylow #sylowtheoremwithidp #internaldirectproduct #sylowpackage #grouptheory #abstractalgebra
Abstract Algebra, Lec 19B, More on Factor (Quotient) Groups, Internal Direct Products
 
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(0:00) Reminder to make sure you truly understand what it means for (left) coset multiplication to be well-defined. (0:58) Outline the rest of the proof that the factor (quotient) group G/H is well-defined. (4:05) The G/Z Theorem and corollaries (including example where G = D6: it turns out that D6/Z(D6) is isomorphic to D3 (and S3) and the fact that if G is a non-Abelian group of order pq, then Z(G) = {e}). (11:08) G/Z(G) is isomorphic to Inn(G). (12:32) Cauchy's Theorem for Abelian groups (a partial converse of Lagrange's Theorem) (a special case of the First Sylow Theorem). (14:24) The proof uses factor groups and induction (one of the most powerful proof techniques in abstract algebra because G/H will be "smaller" than G if H is not {e}). (15:55) Internal direct product (of two subgroups) definition and unique representation consequence (and discuss analog from linear algebra). (21:22) If G = H x K, then, G is isomorphic to H + K and discuss how the isomorphism would be defined. (23:55) General philosophy of external versus internal direct products. (25:05) Another classification fact and corollary (groups of order p^2 are either cyclic or isomorphic to the external direct product of cyclic groups of order p, and are therefore Abelian). (27:19) Notes about proofs to study.
Views: 97 Bill Kinney
301.10 Internal Direct Product and Homomorphism
 
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Class recap just barely introducing homomorphisms
Views: 65 Matthew Salomone
301.8A Direct Products of Groups: Goals
 
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Direct products can be used to "build" larger groups from smaller pieces (each of which is then a subgroup of the result). The immediate goal will be to study how this construction acts on cyclic groups; the larger goal will ultimately be to use these ideas to classify all finite abelian groups.
Views: 131 Matthew Salomone
Abstract Algebra 8.1: External Direct Product
 
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We define the external direct product of a finite number of groups, do some examples, and show why up to isomorphism there are only two groups of order 4.
Views: 85 Patrick Jones
GT6. Centralizers, Normalizers, and Direct Products
 
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Abstract Algebra: We consider further methods of constructing new groups from old. We consider centralizer and normalizer subgroups, which are useful when the group is non-abelian, and direct products. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html
Views: 16617 MathDoctorBob
Direct product representation
 
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Subject: Chemistry Paper: Applications of Molecular Symmetry and Group theory Module: Direct product representation Content Writer:
Views: 991 Vidya-mitra
301.8C Direct Products of Cyclic Groups
 
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A direct product of two cyclic groups is cyclic if and only if the orders of the groups are relatively prime. This has profound implications for how we classify isomorphism classes of finite abelian groups.
Views: 136 Matthew Salomone
Visual Group Theory, Lecture 3.4: Direct products
 
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Visual Group Theory, Lecture 3.4: Direct products There is a natural way to put a group structure on the Cartesian product of two groups. In this lecture, we introduce this concept algebraically, and show several different ways to visualize this, using tools such as Cayley diagrams and multiplication tables. We also look at subgroups and normal subgroups of direct products, and establish a few basic properties. Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/math4120-online.html
Views: 5611 Professor Macauley

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