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. K

    Prove direct product of G1 x G2 is abelian

    Homework Statement Prove that if G1 and G2 are abelian, then the direct product G1 x G2 is abelian. Homework Equations The Attempt at a Solution let G1 and G2 be abelian. Then for a1,a2,b1,b2, we have a1b1=b1a1 and a2b2=b2a2. The direct product is the set of all ordered pairs...
  2. M

    Understanding the Direct Product of Groups: Applying Group Theory Axioms

    How do we know that the cartesian product of any two groups is also a group using the axioms of group theory?
  3. K

    Need Help in Direct Product and Quotient (factor) Group

    Hi All. I have several questions on abstract algebra. Here are my questions and the attempts I had done so far. (1)Let denote Z as the integer.According to theorem, the direct product of Z3 X Z7 = Z21. Hence, is Z4 X Z2 is equal to Z8? Z2X Z2 is equal to Z4? (2)For Z4 ={0,1,2,3 }, and Z2...
  4. K

    Is direct product of subgroups a subgroup?

    Hi.. In the second paragraph of the following paper, there is a statement: "Because the direct product of subgroups is automatically a subgroup.." http://jmp.aip.org/jmapaq/v23/i10/p1747_s1?bypassSSO=1 I don't see how that can be true...you can always take direct product of a subgroup...
  5. T

    Proving D4 Cannot be Expressed as Internal Direct Product

    Homework Statement Prove that D4 (Dihedral group) cannot be expressed as an internal direct product of two proper subgroups. Homework Equations The Attempt at a Solution I know that the only two possible subgroups would be the subgroups of order 4 and 2. I am thinking since D4 is...
  6. X

    Subgroup of external direct product

    I am trying to do the followin 2 problems but not sure if I am doing them correct. Anyone please have a look... 1. In Z40⊕Z30, find two subgroups of order 12. since 12 is the least common multiple of 4 and 3, and 12 is also least common multiple of 4 and 6. take 10 in Z40, and 10...
  7. B

    Properties of the internal direct product of subgroups

    Homework Statement Let H and K be groups and let G = H x K. Recall that both H and K appear as subgroups of G in a natural way. Show that these subgroups H (actually H x {e}) and K (actually {e} x K) have the following properties: a: every element of G is of the form hk for some h in H and k...
  8. N

    Validity of Direct Product Structure of Symmetry Group

    In short, if we consider the group of symmetries of a regular octahedron, we see (or at least, the author of "Groups, Graphs and Trees" saw...) that the group is isomoprhic to Z2\otimesZ2\otimesZ2\otimesS3 - particularly since if we break up the vertices into 3 groups of front-back, top-bottom...
  9. T

    Direct Product Basis for Interacting Systems

    Hi. I found in a book on quantum optics (Vogel, Welsch - Quantum Optics) and also in a lecture script the following statements. A System is composed of two subsystems (say A and B) which interact. So the total Hamiltionian is H_{AB}= H_A + H_B + H_{int}. Nontheless the states of H_{AB}...
  10. M

    Abstract algebra - direct sum and direct product

    Hi everybody, I'm new to absract algebra and I really can not understand different between direct sum and direct product in group theory (specially abelian groups). could does anyone give me a clear example or ... ? thanks
  11. pellman

    Notation question: like a direct product

    I am coming across a notation in a math text whose form and context suggest it is something like a direct product. The symbol is a multiplication X with a vertical line connecting the two left side ends. Can someone identify this symbol please? The context can be seen here...
  12. C

    Proving Associativity of Direct Product of Two Groups

    Suppose you had the following: (A,*) and (B,\nabla) So to prove associativity, since I know that both A and B are groups, their direct product will be a group. Could I do the following ai , bi \in A,B [(a1,b1)(a2,b2)](a3,b3)=(a1,b1)[(a2,b2)(a3,b3)] Since A and B are groups, I...
  13. D

    Isomorphism between groups, direct product, lcm, and gcd

    1. The problem statement, all variables and given/known data Let a,b, be positive integers, and let d=gcd(a,b) and m=lcm(a,b). Show ZaXZb isomorphic to ZdXZm Homework Equations m=lcm(a,b) implies a|m, b|m and if a,b|c then m|c. d=gcd(a,b) implies d|a, d|b and if c|a and c|b then d|c...
  14. C

    When Should Direct Product and Direct Sum Be Used in Group Theory?

    I wasn't sure where to put this since this is under group theory. I am having a little bit trouble understand when to use direct product vs direct sum. One question I have about this is is that if you have two vector spaces that are orthogonal to each other (and example of this might be two...
  15. T

    Find the elements of the direct product

    Homework Statement Compute the direct sum Z_12 (+) U(10) Z_24 is the group Z under addition modulo 12 U(10) is the group Z under multiplication modulo 10 The Attempt at a Solution I have computed direct sums of Z_n groups before: For example: Z_2 (+) Z_3 =...
  16. K

    Why use direct product construction in quantum mechanics?

    For a system consists of two particle, says two spins, the wavefunction for that system is a direct product of each individual states. And the corresponding operator is also a direct product of each individual operators. I know this is a procedure to construct such a space, but can anybody told...
  17. F

    Ideals of direct product of rings are direct product of respective ideals?

    I want to answer this question: Find all the ideals of the direct product of rings R \times S. (I think this means show that the ideals are I \times J where I, J are ideals of R, S, respectively.) I think the problem is that I don't know how to show that any ideal of R \times S is of the...
  18. J

    Decomposition of direct product into symmetric/antisymmetric parts

    Can anyone explain to me why the 3-rep of SU(3) gives 3\otimes 3 = \overline{3}\oplus 6 whereas for the 5 of SU(5) 5\otimes 5 = 10\oplus 15? I thought the general pattern was N \otimes N = \overline{\frac{1}{2}N(N-1)}\oplus \frac{1}{2}N(N+1) but this second example seems to...
  19. P

    Is H1xH2 a Subgroup of G1 X G2?

    Homework Statement Let G1, G2 be groups with subgroups H1,H2. Show that [{x1,x2) | x1 element of H1, x2 element of H2} is a subgroup of the direct product of G1 X G2 The Attempt at a Solution I'm not sure how to begin solving this problem.
  20. F

    How Do Elements of Direct Product Groups Commute in Particle Physics?

    From Gauge Theory of particle physics, Cheng and Li I don't understand the flollowing: "Given any two groups G={g1,..} H= {h1,h2,...} if the g's commute with the h's we can define a direct product group G x H={g_ih_j} with the multplication law: g_kh_l . g_mh_n = g_kh_m . h_lh_n Examples...
  21. P

    Is G isomorphic to a subgroup of G/M\times G/N?

    Homework Statement Let M and N be normal subgroups of G, and suppose that the identity is the only element in both M and N. Prove that G is isomorphic to a subgroup of the product G/M\times G/N Homework Equations Up until now, we've dealt with isomorphism, homomorphisms, automorphisms...
  22. M

    Semi-Direct Product: Homomorphisms, Generators, Relations & Isomorphism Types

    thnx for helping on the previous post. heres the next one, as usual any hints on how to approach the question would be greatly appreciated. i will attempt the questions with the hints :) (1) describe explicitly all homomorphisms \varphi : C_4 \rightarrow Aut(C_5) (2) For each...
  23. T

    Isomorphism and direct product of groups

    Just wondering if there is a general way of showing that (Z, .)n isomorphic to Zm X Zp with the obvious requirement that both groups have the same order?
  24. radou

    What is the Definition of a Weak Direct Product in Group Theory?

    Well, I have trouble understanding the definition of the weak product of a family of groups, which states that it is the set of all f \in \prod_{i \in I} G_{i} such that f(i) = e_{i} \in G_{i}, for all but a finite number of i from I (Gi is a family of groups indexed by the set I). The bolded...
  25. B

    Abstract Algebra - Direct Product Question

    I'm supposed to find a non-trivial group G such that G is isomorphic to G x G. I know G must be infinite, since if G had order n, then G x G would have order n^2. So, after some thought, I came up with the following. Z is isomorphic to Z x Z. My reasoning is similar to the oft-seen proof...
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