- #1

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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

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- Thread starter markoX
- Start date

- #1

- 28

- 0

I'm new to absract algebra and I really can not understand different between

could does any one give me a clear example or ... ?

thanks

- #2

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- #3

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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?

- #4

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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?

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?

- #5

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how do these two objects can be same in Z-modules group?

- #6

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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].

- #7

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how do these two objects can be same in Z-modules group?

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.

- #8

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- #9

Fredrik

Staff Emeritus

Science Advisor

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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.

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:

- #10

Landau

Science Advisor

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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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