# Homework Help: First Isomorphism Theorem

1. Nov 21, 2006

### 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?

2. Nov 21, 2006

### 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.

Last edited: Nov 21, 2006
3. Nov 21, 2006

### Hurkyl

Staff Emeritus
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...)