Question

9. Suppose n : (Z4 ⓇZ3) + (Z2 Z3) is defined by n(a,b) = (a mod 2, 0). This is a homomorphism. (You can believe me about that, and you don't need to check.) Write down the elements of the kernel of n.

Answer 1

Ker($\eta$) = { (a,b) in Z₄ ® Z₃ : $\eta$ (a,b) = (0,0) }

So, $\eta$ (a,b) = (a mod 2, 0) = (0,0)

So, a (mod 2) = 0

So, 2 divides a

So, a = 0, 2

So, the kernel is,

ker($\eta$) = { (a,b) in Z₄ ® Z₃ : a = 0, 2 and b is in Z₃ }

= { (0,0), (0,1), (0,2), (2,0), (2,1), (2,2) }