My understanding was that the product of two groups A and B will yield a group C for which the dimension of C is dim(A)*dim(B). Now however, the author I'm reading defines the group product multiplication as: (a1, b1) * (a2, b2) = (a1*a2, b1*b2), for a1,a2 in A and b1, b2 in B. Does this give the same result for the dimension?? Lets take the example with A and B corresponding to the real numbers with addition as a group operation. This gives: (x1, y1) + (x2, y2) = (x1+x2, y1+y2).... It looks like we have created the two dimensional plane! But dim(R)*dim(R) = 1 * 1 = 1. C should have dimension 1 not 2. :/ Please help me!