# Thread: Needs Counterexample for homomorphisms

1. ## Needs Counterexample for homomorphisms

Let A, B be groups and A' and B' be normal subgroups of A and B respectively. Let f: A --> B be a homomorphism with f(A') being a subgroup of B'. There is a well-defined homomorphism g: A/A' -----> B/B' defined by g: aA' ---> f(a)B'

Find an example in which f is injective, but g is not injective.

I've proven that g is a well-defined homomorphism and that if f is surjective, then g is surjective; but don't really know how to go with this.

2. Originally Posted by playa007
Let A, B be groups and A' and B' be normal subgroups of A and B respectively. Let f: A --> B be a homomorphism with f(A') being a subgroup of B'. There is a well-defined homomorphism g: A/A' -----> B/B' defined by g: aA' ---> f(a)B'

Find an example in which f is injective, but g is not injective.
How about $A=B=\mathbb{Z}$, f = identity map, A' = {multiples of 4}, B' = {multiples of 2}?