# Math Help - homomorphism

1. ## homomorphism

G = Z_4 x Z_4

prove that the map theta: Z_4 x Z_4 --> Z_4 defined by theta(([a],[b])) = [b] is a homomorphism ONTO Z_4.

what is the kernel of theta? Show that the quotient group G/<([1],[0])> is isomorphic to Z_4

2. Originally Posted by tas10
G = Z_4 x Z_4

prove that the map theta: Z_4 x Z_4 --> Z_4 defined by theta(([a],[b])) = [b] is a homomorphism ONTO Z_4.

what is the kernel of theta? Show that the quotient group G/<([1],[0])> is isomorphic to Z_4
Where's the work? It's just as easy to prove in general the projection $\pi:G\times H\to Gg,h)\mapsto g" alt="\pi:G\times H\to Gg,h)\mapsto g" /> is an epimorphism.