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
http://amzn.to/2oOBd5S
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: 50705
Socratica

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

Views: 1294
Reesha Education

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

Views: 7345
Ben1994

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: 6241
CSIR NET GATE CHEMISTRY

In this video we discuss the construction of product groups.

Views: 3078
Ben1994

We define the product of two groups and prove that the operation on the product does indeed define a group structure.

Views: 5798
Tim Hodges

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: 1012
Mathematical Science

Cyclic groups are the building blocks of abelian groups. There are finite and infinite cyclic groups. In this video we will define cyclic groups, give a list of all cyclic groups, talk about the name “cyclic,” and see why they are so essential in abstract algebra.
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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
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Be sure to subscribe so you don't miss new lessons from Socratica:
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison

Views: 118649
Socratica

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

Views: 3510
LadislauFernandes

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: 16139
MathDoctorBob

Direct Product of Normal Subgroups is Normal Proof. In this video we prove that if A is a normal subgroup of G and B is a normal subgroup of H, then A x B is a normal subgroup of G x H.

Views: 1535
The Math Sorcerer

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: 128
Vivek Kumar Jain

Views: 80
Killian O'Brien

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

Views: 868
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: 29213
scalenescott

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

Views: 1214
Mathematical Science

The external direct product of A and C is always a solution to the "product problem" of short exact sequences. Here we see how.

Views: 33
Matthew Salomone

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: 359
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: 68
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: 135
Vivek Kumar Jain

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

Views: 930
Vidya-mitra

Group Theory 32, External Direct Products

Views: 7026
LadislauFernandes

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: 5262
Professor Macauley

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: 3174
The Math Sorcerer

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

Views: 2123
kamil nawaz

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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Views: 3982
TMP Chem

Direct Products of Finite Cyclic Groups Video 2. How to determine if the direct product of finite cyclic groups is cyclic. Better examples than the first video.

Views: 2022
The Math Sorcerer

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: 269
Mathematical Science

When we multiply the elements of two subgroups of G together, how many elements are there in the resulting set? This product formula theorem is a valuable element in classifying groups whose order is twice a prime, and is also an illustration of a more general product construction that we'll use to understand how larger groups are put together from smaller.

Views: 54
Matthew Salomone

(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: 410
Bill Kinney

Topic Covered : Product of Subgroup and Theorem and its examples. Multiplication of Subgroup
Facebook page ..
https://www.facebook.com/Math.MentorJi/
Graduate Math app :https://goo.gl/vo2Tj2
thoerem related subgroup : https://youtu.be/cfT3ZFmfNLI
Subgroup and examples = https://youtu.be/H7CKR1Nevnw
(a+b)^2 =https://youtu.be/5i5yL2BCwpc
permuatation group theory :https://youtu.be/-VvUsxsujyc
Examples of singularity :
https://youtu.be/cgsB8Z5WSPk
Riemann Sum : https://youtu.be/Z3Ecy2Zwukw
Riemann Sum problems https://youtu.be/LKuZreMPiRQ
infimum and supremum https://youtu.be/mK6NZznoZeg
Dirichlet and able test https://youtu.be/WyoMpdh7f0c
uniform convergence : https://youtu.be/_WWsMl0_9BI
MN Test of uniform : https://youtu.be/r5yec-FtlUE
pointwise convergence:http://youtu.be/o_0YjNo_v64
Cauchy integral Formula :http://youtu.be/LEJBT0nLngM
complex integration : http://youtu.be/s2wPryo_Hfs
Limit pt .of infimum & supremue :http://youtu.be/zIn8CTcX-6A
comparsion test(convergence) : http://youtu.be/02IncEDug2Y
Cauchy all theorm :http://youtu.be/G5ZTzjN8KQA
Cauchy sequence with example :http://youtu.be/B-7cUVXSZeI
Cauchy nth root test :http://youtu.be/AOPIZsR4JkU
Basics Of Sequence And Series :http://youtu.be/IZgNfFc481M
Convergence sequence : http://youtu.be/c3Il3eEPvF0
Power series radius,domain convergent:https://youtu.be/C8Bw-gFC1Gg

Views: 2156
Math Mentor

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

Views: 2445
Ben1994

We look at an easy way to determine whether the direct product of cyclic groups is itself cyclic or not.

Views: 12
Patrick Jones

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: 32
Patrick Jones

Group Theory 33, Properties of External Direct products

Views: 5408
LadislauFernandes

(0:00) Lecture plan. (0:29) Exam 2 next class period. (1:35) Internal direct product of two normal subgroups and the proof that it is isomorphic to the corresponding external direct product. (18:42) Definition of a group homomorphism and the kernel of a group homomorphism (with comments linear transformations and null spaces and about preimage (inverse image) notation). (22:15) Kernels are normal subgroups and cosets of the kernel are preimages of individual elements in the image ("range"), and a verification. (29:19) First Isomorphism Theorem and the idea of its proof. (35:51) Normal subgroups are kernels.

Views: 155
Bill Kinney