Does the First Isomorphism Theorem apply in this case?

[JasonJo]

prove that there does not exist a homomorphism from G:= (integers modulo 8 direct product integers modulo 2) to H:= (intergers modulo 4 direct product integers modulo 4).

Pf:
i tried this route, assume that there is such a homomorphism. then by first isomorphism theorem, G/ker phi is isomorphic to phi(G) but what would the kernel of phi have to be in this case?

 

[0rthodontist]

Z8 x Z2 => Z4 x Z4??
There is a homomorphism. For example phi(a, b) = (0, 2b) describes a nontrivial homomorphism, unless I've gone blind.

 

[Hurkyl]

Hrm. Maybe he means to consider them as rings, rather than as additive groups?

Ring homomorphisms must preserve the identity. (As well as all integer multiples of the identity...)