# homomorphism

• May 9th 2010, 07:44 PM
tas10
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
• May 9th 2010, 07:46 PM
Drexel28
Quote:

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 $\pi:G\times H\to G:(g,h)\mapsto g$ is an epimorphism.