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.
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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
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Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison

Views: 56101
Socratica

Definition and theorems related to External Direct Product valuable for exam as well

Views: 2391
Reesha Education

Group Theory 35, Internal Direct Products
http://groupprops.subwiki.org/wiki/Internal_direct_product

Views: 3799
LadislauFernandes

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

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.

Views: 8305
CSIR NET GATE CHEMISTRY

In this video we discuss the definition of the internal and external direct products.

Views: 7917
Ben1994

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

#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
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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.
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Views: 2
Audioversity

Group Theory 33, Properties of External Direct products

Views: 6104
LadislauFernandes

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

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

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

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

Here I have discuss about External Direct Product(Definition with Example)

Views: 62
point Creation

(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

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

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

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

Subject: Chemistry
Paper: Applications of molecular symmetry and group theory
Module: Direct product representation
Content writer: Prof B S Garg

Views: 982
Vidya-mitra

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

Easy way to find the order of elements in direct product of groups
#orderofelementsofcartesiangroup
#orderofelemets
#directproductofgroups
#cartesianproductofgroups
#grouptheory
#abstractalgebra

Views: 1641
Mathematical Science

(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

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
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Screen Capture: Corel Visual Studio Pro X8

Views: 4290
TMP Chem

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.
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Author-Info: derivative work: Pbroks13 (talk) Cyclic_group.png: Jakob.scholbach
Image Source: https://en.wikipedia.org/wiki/File:Cyclic_group.svg
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https://www.youtube.com/watch?v=szHP0cnyq-o

Views: 390
WikiAudio

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

Views: 7275
LadislauFernandes

INTERNAL DIRECT PRODCUT FIRST LECTURE BY SIR KAMIL NAWAZ GROUP THEORY ADVANCE ALGEBRA MSC MATHEMATICS DEFINITION OF INTERNAL DIRECT PRODDUCT

Views: 2446
kamil nawaz

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

Views: 644
Mathematical Science

(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

Class recap just barely introducing homomorphisms

Views: 65
Matthew Salomone

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

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

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

Subject: Chemistry
Paper: Applications of Molecular Symmetry and Group theory
Module: Direct product representation
Content Writer:

Views: 991
Vidya-mitra

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
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