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Abstract algebra - direct sum and direct product

  1. Sep 1, 2009 #1
    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 any one give me a clear example or ... ?
    thanks
     
  2. jcsd
  3. Sep 2, 2009 #2
    I think that direct sum refers to modules over a ring. One takes a direct product of abelian groups to get another abelian group. But if you view an abelian group as a Z-module then the direct product is the direct sum of Z-modules.
     
  4. Sep 2, 2009 #3
    thanks for reply,
    Do you mean direct product and direct sum are the same for Z-modules?
    but how do their definition are different for two matrices A and B as you know?
     
  5. Sep 2, 2009 #4
    I think it is a direct product if you view the groups as groups, a direct sum if you view them as Z-modules. They are not really the same because they are being view as different types of objects.

    I don't understand you matrix question. Can you explain it more?
     
  6. Sep 2, 2009 #5
    My second question is not related to group theory, suppose we have two matrices A and B. The direct product of these two matrices is A * B ( which is tensor product ) but the direct sum is something else.
    how do these two objects can be same in Z-modules group?
     
  7. Sep 2, 2009 #6
    Each of these (direct sum, direct product) is the solution of a certain universal mapping problem. In the case of abelian groups, the resulting groups are isomorphic, but not the resulting maps.

    For the "direct sum": given two abelian groups [itex]A, B[/itex] you get the direct sum group [itex]A \times B[/itex] and two embeddings, [itex]i_1 \colon A \to A \times B[/itex] and [itex]i_2 \colon B \to A \times B[/itex].

    For the "direct product": given two abelian groups [itex]A, B[/itex] you get the direct product group [itex]A \times B[/itex] and two projections, [itex]p_1 \colon A \times B \to A[/itex] and [itex]p_2 \colon A \times B \to B[/itex].
     
  8. Sep 2, 2009 #7
    I have never heard the tensor product called a direct product. If that is what your book says then this to me is non-standard terminology.

    The direct sum of two matrices(linear maps) act on the direct sum of the two vector spaces - the tensor product acts on the tensor product of the vector spaces. If the 2 vector spaces have dimensions m and n then the dirct sum has dimension m + n , the tensor product has dimension mxn.
     
  9. Mar 14, 2010 #8
    In mathematics, the direct sum of groups: \Pi_{i\in I} G_i is the set of all "sequences" (x_i)_{i\in I} such that x_i\in G_i for all i\in I. The direct sum \bigoplus_{i\in I}G_i is the subset of the direct product consisting of the sequences with all except finitely many terms equal to the identities of the relevant groups. Thus, if I is finite the direct product is the same as the the direct sum .
     
  10. Mar 14, 2010 #9

    Fredrik

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    Sounds like most of you are making it more complicated than it needs to be. Either that or I have misunderstood something. I guess I'll find out now. Here's how I would define those terms:

    If G and H are groups, then the direct product of G and H is the Cartesian product G×H with the multiplication operation defined by (g,h)(g',h')=(gg',hh').

    The direct sum is exactly the same thing. The only difference is that when we're dealing with Abelian groups, we often use the notation g+g' instead of gg'. When we do, the definition of the "multiplication" operation on G×H is written as (g,h)+(g',h')=(g+g',h+h') instead of as above. It's still the same definition, but now we call the operation "addition" instead of "multiplication".

    ...and I see now that this thread is more than 6 months old.
     
    Last edited: Mar 14, 2010
  11. Mar 14, 2010 #10

    Landau

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    @Fredrik: as Nessy said, for finite products/sums, there is no difference between direct sum and direct product. You are talking about two (=finitely many) groups G and H, so you're right.

    The difference comes up when dealing with infinite products and sums, i.e. [tex]\prod_{i\in I}G_i[/tex] and [tex]\bigoplus_{i\in I}G_i[/tex], where I is an infinite index set.

    \\edit: also, see here.
     
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