# Direct product/sum of groups

## Main Question or Discussion Point

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.

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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.