Direct Sum/Product of Groups Clarification?

[shallumstuart]

I have a pretty basic question about direct sum/product of groups.

Say you were given the group (Z4 x Z2, +mod2). Now I know that Z4 x Z2 is given by { (0,0), (0,1), (1,0), (1,1), (2,0), (2,1), (3,0), (3,1) }. So now if you were going to add together two of the elements using the binary operation +mod2, e.g. doing (1,1) +mod2 (2,1). Does this give you:

(1,1) +mod2 (2,1) = (1+2,1+1) = (3,2) = (1,0)?
I'm pretty sure that this is correct, but I thought another possibility might have been that you add the first two elements in mod4 and the second two in mod 2

e.g. (1,1) + (2,1) = (3,2) = (3,0).

Help clarifying would be super.

 

[eok20]

The definition of the direct product of two groups G and H is that you multiply (or add in the abelian case) componentwise using the operation in the given component. So in Z4 x Z2 the operation is the second one you did, e.g. (1,1) + (2,1) = (3,0). Of course to get the first thing you had (e.g. (1,1) + (2,1) = (1,0)) you could redefine the operation in Z4 to be addition mod 4, but then you would really have Z2 (so the direct product would be Z2 x Z2). For example with this operation on Z4 we are forced for 3 and 1 to be the same element since 3 - 1 = 2 = 0.

 

[shallumstuart]

thanks that's helpful.

although i still don't quite get the meaning of the +mod2 part. When the group is defined as (Z4 x Z2, +mod2), doesn't the +mod2 tell you the binary operation by which you combine elements of the set? if that's the case, when you add componentwise how come you are adding the first two together in mod4 but the second two in mod2?

 

[shallumstuart]

never mind, i don't actually think it makes any sense to say (Z4 x Z2, +mod2)

 

[blue_raver22]

I can confirm that your understanding of the direct sum/product of groups is correct. When adding two elements in a direct sum/product, we use the binary operation defined for each individual group. In this case, the binary operation for Z4 is addition mod 4 and the binary operation for Z2 is addition mod 2. Therefore, (1,1) +mod2 (2,1) would be (1+2,1+1) = (3,2) = (1,0) in Z4 x Z2.

Your second possibility of adding the first two elements in mod4 and the second two in mod2 is not correct. In a direct sum/product, the elements are added individually according to their group's binary operation. So, (1,1) + (2,1) would not be (3,2) = (3,0), but rather (1+2,1+1) = (3,2) = (1,0) in Z4 x Z2.

I hope this clarifies any confusion and helps you better understand direct sum/product of groups. Keep up the good work!