Abstract algebra - direct sum and direct product

[markoX]

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

 

[wofsy]

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.

 

[markoX]

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?

 

[wofsy]

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?

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?

 

[markoX]

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?

 

[g_edgar]

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 $A, B$ you get the direct sum group $A \times B$ and two embeddings, $i_1 \colon A \to A \times B$ and $i_2 \colon B \to A \times B$.

For the "direct product": given two abelian groups $A, B$ you get the direct product group $A \times B$ and two projections, $p_1 \colon A \times B \to A$ and $p_2 \colon A \times B \to B$.

 

[wofsy]

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?

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.

 

[Nessy]

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 .

 

[Fredrik]

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.

 

[Landau]

@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. $$\prod_{i\in I}G_i$$ and $$\bigoplus_{i\in I}G_i$$, where I is an infinite index set.

\\edit: also, see here.