What is Direct product: Definition and 75 Discussions

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.
Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance.There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.

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  1. C

    Direct product of two semi-direct products

    After finding the number of elements for this group, how do I extend the argument to $$p,q\equiv1\left(mod\ 3\right)$$, where $$G=(C_p:C_3\ )\times(C_q:C_3\ )$$Any help appreciated.
  2. J

    Is the Direct Product of Groups Associative and Have an Identity Element?

    So I that I need to prove the axioms: associativity, existence of the identity element, and existence of the right inverse. For associativity I know that the binary operations of G and H have to already be associative, and the elements of G X H are made up of these binary operations, so...
  3. F

    U(n) as an External Direct Product

    Homework Statement EDIT: ##U(n)## is the set of relatively prime numbers less than ##n## and ##U_k(n) = \lbrace x \epsilon U(n) : x \equiv 1 (\operatorname{mod} k) \rbrace##. I'm trying to finish the proof of this statement(s): Suppose ##s## and ##t## are relatively prime. Then ##U(st) \approx...
  4. Mr Davis 97

    Showing that Q_8 can't be written as a direct product

    Homework Statement Prove that the quaternion group ##Q_8## is not isomorphic to a semi-direct product ##H\rtimes_\rho K## for non-trivial groups ##H## and ##K##. Homework EquationsThe Attempt at a Solution My idea is to look at the subgroups of ##Q_8## and to show that the intersection of any...
  5. B

    Is N Always in the Center of G or Does It Intersect H or K Nontrivially?

    Homework Statement Let ##H, K, N## be nontrivial normal subgroups of a group ##G## and suppose that ##G = H \times K##. Prove that ##N## is in the center of ##G## or ##N## intersects one of ##H,K## nontrivially Homework EquationsThe Attempt at a Solution I presume that ##G = H \times K##...
  6. T

    Finding the subgroups of direct product group

    Homework Statement What are the subgroups of Z2 x Z2 x Z2? Homework Equations Hint: There are 16 subgroups. The Attempt at a Solution So far I only manage to get 15 and I am not even sure if these are correct. My answer: $$(0,0,0) , (Z_2,Z_2,Z_2), (1,1,1), (0,0,1), (0,1,0), (1,0,0), (0,1,1)...
  7. B

    Groups that cannot be the direct product of subgroups

    Homework Statement I am trying to show that neither ##Z_{p^n}## nor ##\mathbb{Z}## can be written as any family of its proper subgroups. Homework EquationsThe Attempt at a Solution First, I believe this solution (http://www.auburn.edu/~huanghu/math7310/7310-hw2-answer.pdf see problem 6) is...
  8. Mr Davis 97

    The order of an element of a direct product of groups

    Homework Statement Let ##A## and ##B## be finite groups, and ##A \times B## be their direct product. Given that ##(a,1)## and ##(1,b)## commute, and that ##(a,1)^n = (a^n,1)## and ##(1,b)^n = (1,b^n)## for all a and b, show that the order of ##(a,b)## is the least common multiple of the orders...
  9. F

    I Direct product of a symmetry group with itself

    In group theory, what is the direct product of a symmetry group with itself? Say T*T or O*O?
  10. Mr Davis 97

    Abelian group as a direct product of cyclic groups

    Homework Statement Consider G = {1, 8, 12, 14, 18, 21, 27, 31, 34, 38, 44, 47, 51, 53, 57, 64} with the operation being multiplication mod 65. By the classification of finite abelian groups, this is isomorphic to a direct product of cyclic groups. Which direct product? Homework EquationsThe...
  11. S

    I Properties of Direct Product of Half Open and Open Intervals

    The 2-D plane is usually constructed as "ℝxℝ" and ℝ is both open and closed. My question is, what is the direct product of a half open and an open interval? Is it also open or half open?
  12. I

    I Interpretation of direct product of Hilbert spaces

    Dear all, I know how to interpret a vector, inner product etcetera in one Hilbert space. However, I can not get my head around how the direct product of two (or more) Hilbert spaces can be interpreted. For instance, the Hilbert space ##W## of a larger system is spanned by the direct product of...
  13. G

    I Why the tensor product (historical question)?

    Hi. Why did the founding fathers of QM know that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces and not a direct product, where no entanglement would emerge? I mean today we can verify entanglement experimentally, but this became technologically...
  14. S

    I Difference between direct sum and direct product

    Hello! I am reading something about applications of group theory in quantum mechanics and I got confused about the difference between direct sum and direct product. In many places I found that they mean the same thing. However, the ways I found them defined in the book I read from, seem to be...
  15. Math Amateur

    MHB Paul E Bland's "Direct Product of Modules" Definition - Category-Oriented

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I am trying to fully understand Bland's definition of a direct product ... and to understand the motivation for the definition ... and the implications of...
  16. DeldotB

    Show a group is a semi direct product

    Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...
  17. DeldotB

    Why a group is not a direct or semi direct product

    Homework Statement Good day all! (p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out) So I was wondering if my understanding of this is correct: The Question asks: "\mathbb{Z}_4 has a subgroup is isomorphic to...
  18. D

    Direct Product vs Tensor Product

    Hi, I am working through a textbook on general relativity and have come across the statement: "A general (2 0) tensor K, in n dimensions, cannot be written as a direct product of two vectors, A and B, but can be expressed as a sum of many direct products." Can someone explain to me how this...
  19. S

    Direct product representation of a function?

    When do functions have representations as a "direct product"? For example, If I have a function f(x) given by the ordered pairs: \{(1,6),(2,4),(3,5),(4,2),(5,3),(6,1) \} We could (arbitrarily) declare that integers in certain sets have certain "properties": \{ 1,3\} have property A...
  20. Safinaz

    Direct product of two representations

    Hi their, It's a group theory question .. it's known that ## 10 \otimes 5^* = 45 \oplus 5, ## Make the direct product by components: ##[ (1,1)^{ab}_{1} \oplus (3,2)^{ib}_{1/6} \oplus (3^*,1)^{ij}_{-2/3} ] \otimes [ (1,2)_{ c~-1/2} \oplus (3^*,1)_{ k~1/3} ] = (1,2)^{ab}_{ c~1/2} \oplus...
  21. D

    Direct Product of Two Spin-Up Vectors: What Is It?

    Hi I have just started looking at direct products and came across the following which i don't understand : the direct product of two spin -up vectors = | 1 > which is in a bigger vector space I don't understand how the direct product is | 1 > ? and in this case is it always a bigger vector...
  22. J

    Are \bigoplus and \times interchangeable in direct sum and direct product?

    Under what conditions are the symbols \bigoplus and \times intechangangable?
  23. B

    Is the Direct Product of Two Finite Cyclic Groups Cyclic

    Hello everyone, I was wondering if the following claim is true: Let ##G_1## and ##G_2## be finite cyclic groups with generators ##g_1## and ##g_2##, respectively. The group formed by the direct product ##G_1 \times G_2## is cyclic and its generator is ##(g_1,g_2)##. I am not certain that it...
  24. S

    MHB Problem about a group with two inner direct product representations

    The problem: Suppose G is Abelian with two representations as the internal direct product of subgroups: G=HxK1, G=HxK2. Assume K1 is a subset of K2 and show K1=K2. My attempted solution: I took the element (e_H, k_2), where e_H is the identity element of H and k_2 is an arbitrary element in K2...
  25. A

    Commuting operators and Direct product spaces

    Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? As I'm not being able to precisely phrase my doubt, consider this example: Hilbert space of a two dimensional particle is the direct product of...
  26. Math Amateur

    MHB Definition of Direct Product Using UMP - Including Infinite Case

    I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the general UMP-based definition of external and internal direct products ... ... On page 63, Knapp defines...
  27. G

    Direct product of space and spin in hadrons

    The uuu hadron doesn't violate Pauli's exclusion principle presumably because there is color. But even without color, can't the uuu exist if spatial wavefunctions are different? Suppose one u quark is located at r1, another at r2, and another at r3, and say that all three u quarks have spin up...
  28. Deveno

    MHB The Universal Property of the Direct Product in Groups

    With groups, one often seeks to create larger groups out of smaller groups, or the reverse: break down large groups into easier-to-understand pieces. One construction often employed in this regard is the direct product. The normal way this is done is like so: The direct product of two groups...
  29. T

    Direct Sum and Direct Product: Understanding the Differences in Vector Spaces

    The definition (taken from Robert Gilmore's: Lie groups, Lie algebras, and some of their applications): We have two vector spaces V_1 and V_2 with bases \{e_i\} and \{f_i\}. A basis for the direct product space V_1\otimes V_2 can be taken as \{e_i\otimes f_j\}. So an element w of this space...
  30. K

    MHB Why a group is not isomorphic to a direct product of groups

    I would like to know why $M_n$ $\not\cong$ $O_n$ x $T_n$, where $M_n$ is the group of isometries of $\mathbb R^n$, $O_n$ is the group of orthogonal matrices, and $T_n$ is the group of translations in $\mathbb R^n$. **My attempt:** Can I show that one side is abelian, while the other group is...
  31. L

    Do Tensor Product Properties Hold in Infinite Dimensional Hilbert Spaces?

    Is this correct in infinite dimensional Hilbert spaces? ## (\hat{A}_1 \otimes \hat{A}_2)^{-1}=\hat{A}^{-1}_1 \otimes \hat{A}^{-1}_2 ## ## (\hat{A}_1 \otimes \hat{A}_2)^{\dagger}=\hat{A}^{\dagger}_1 \otimes \hat{A}^{\dagger}_2 ## ## (\hat{A}_1 +\hat{A}_2) \otimes \hat{A}_3=(\hat{A}_1 \otimes...
  32. L

    S^2 × S^2...×S^2×S^2 is a Direct Product of S^2 - Hypertorus

    http://en.wikipedia.org/wiki/Torus ##S^1 \times S^1... \times S^1 \times S^1 ## is hypertorus. And what is ##S^2 \times S^2... \times S^2 \times S^2 ##?
  33. P

    Difference between tensor product and direct product?

    Hi, I have been learning about tensor products from Dummit and Foote's Abstract Algebra and I'm a little confused. I understand the construction of going to the larger free group and "modding out" by the relations that will eventually end up giving us module structure. But just in the...
  34. caffeinemachine

    MHB Direct product of abelian groups. Isomorphism.

    Let $A,B,C$ be finite abelian groups. Assume that $A\times B\cong A\times C$. Show that $B\cong C$. I observed that $(A\times B)/(A\times\{e\})\cong B$ and $(A\times C)/(A\times\{e\})\cong C$. So I need to show that $(A\times B)/(A\times\{e\})\cong (A\times C)/(A\times\{e\})$. Let...
  35. B

    Direct Product of Groups: Subgroup Realization and Diagonal Subgroup

    I was reading on wikipedia on direct product of groups because I wanted find out if every subgroup of G \times H is realized as a direct product of subgroups of G and H. Apparently it is not, because the diagonal subgroup in G \times G disproves this. I'm a little confused, because I thought...
  36. P

    How to evaluate the commutator of direct product?

    assuming that A B C D are all n\times n operators how to evaluate the commutator of direct product? [A\otimes B, C\otimes D]
  37. K

    Direct product between vectorspaces

    Hello! I am currently reading the analysis of tensors and have now encountered the tensorproduct, \otimes . I am wondering about the statement that every vector in: V \otimes W (with the basis (v_i) and (w_i)) can be written as a linear combination of the basis: v_i \otimes w_i , but not in...
  38. R

    Direct product of faithful representations into direct sum

    Direct product of two irreducible representations of a finite group can be decomposed into a direct sum of irreducible representations. So, starting from a single faithful irreducible representation, is it possible generate every other irreducible representation by successively taking direct...
  39. B

    Isomorphic direct product cyclic groups

    Help! For p prime I need to show that C_{p^2} \ncong C_p \times C_p where C_p is the cyclic group of order p. But I've realized I don't actually understand how a group with single elements can be isomorphic to a group with ordered pairs! Any hints to get me started?
  40. H

    Direct product of two groups with different n-spaces

    how does one evaluate the direct product between a group G with components that are say 2-tuple and a group H with components that are just 1-tuple?
  41. E

    Decomposing the direct sum as direct product

    This is a basic question in angular momentum in quantum mechanics that I am studying. I know that \frac{1}{2}\otimes \frac{1}{2} = 1\oplus 0 What would be a strategy to proving the general statement for spin representations j\otimes s =\bigoplus_{l=|s-j|}^{|s+j|} l
  42. A

    Simple Q about direct product representation of a group

    (At least, I think it's simple.) Disclaimer: I'm approaching this subject from the vantage point of a chemist, so be careful with how much lingo/jargon/rigor you lay on me :redface: The claim is that if you have two representations of a group, \Gamma_1 and \Gamma_2, with bases \{ f_i \} and \{...
  43. M

    DIHEDRAL GROUP - Internal Direct Product

    I have to prove that D4 cannot be the internal direct product of two of its proper subgroups.Please help.
  44. M

    Comparing Elements of Order 4 in External Direct Products

    Homework Statement Explain why external direct products z8 + z4 and z80000000+ z4000000 have same number of elements of order 4? Homework Equations The Attempt at a Solution Z 8 = { 0, 1, 2, 3, 5, 6, 7, } , order of elements : 0 =1, 1=8, 2=4 , 3=8, 4=2, 5=8, 6=4, 7=8. Z 4= {...
  45. M

    External direct product- homomorphism

    Theorem 8.3 in Gallian's Contemp. Abst Alg says with (s, t) = 1 the group U(st) is isomorphic to the external direct product of U(s) and U(t) that is, to U(s) (+) U(t) U(n) is the group of positive integers less than n and relatively prime to n with the group operation...
  46. T

    Subgroups/Internal Direct Product

    Homework Statement The Attempt at a Solution < denotes a subgroup and \triangleleft denotes a normal subgroup throughout. Can anyone tell me what I've done right/wrong? I've posted all my working below: To prove that A<G, I can say that: A (2n+1) \times (2n+1) matrix is...
  47. T

    Groups - Internal Direct Product

    Homework Statement [PLAIN]http://img689.imageshack.us/img689/3047/directproduct.png < denotes a subgroup. \triangleleft denotes a normal subgroup. The Attempt at a Solution Have I done (a) correctly? 0 \in A so A \neq \emptyset If a=x+ix and b=y+iy then ab^{-1} = x-y + ix...
  48. S

    Direct product of a subset of a group

    Homework Statement http://math.uchicago.edu/~dc/teaching/254/254_problem_set_05.pdf" Homework Equations H a subgroup of G implies HH = H, where HH is the direct product.The Attempt at a Solution For the forward direction, don't we just have |A| = 0 or 1? I think |A| = 0 should be excluded...
  49. K

    Show HK are subgroups of a direct product G, HK=KH=G

    Homework Statement Let G1 and G2 be groups and let G be the direct product G1 x G2. Let H={(x1,x2) in G1 x G2 such that x2=e} and let K={(x1,x2) in G1 x G2 such that x1=e} a) Show H and K are subgroups of G b) Show HK=KH=G c) Show that H intersect K={(e,e)} Homework Equations...
  50. K

    Showing something is a subgroup of the direct product

    Homework Statement Let G1 and G2 be groups, with subgroups H1 and H2 respectively. Show that {(x1,x2) such that x1 is in H1, x2 is in H2} is a subgroup of the direct product G1 x G2 Homework Equations The Attempt at a Solution let G1, G2 be groups with H1, H2 subgroups. Let...