homomorphic groups without isomorphism

Question

I am just trying to think about if the homomorphism $\delta$ between two groups can be undefined for some elements such as the id map from $\mathbb{Z} \; \mapsto \; \mathbb{N}$ thanks.. well because, I want an example without isomorphism between $\mathbf{G}_{ \; 1} \; / \; ker \; ( \; \delta \; )$ and $\mathbf{G}_{\; 2}$. That is all....

Answer 1

Let $f \colon G \rightarrow H$ be a homomorphism of groups. By the first isomorphism theorem you will always get an isomorphism $G/\text{ker}(f) \cong \text{im}(f)$, but in general you will not have $G/\text{ker}(f) \cong H$. An easy example would be an inclusion of finite groups $G \subset H$, where $G \neq H$. Then you will always have $G/\text{ker}(i) \not \cong H$ due to cardinality reasons. This example obviously also works with embeddings in the same way or with a finite subgroup of an infinite group.

As a concrete example, just consider the inclusion $\lbrace 0 \rbrace \subset \mathbb{Z}$.

In case that this does not answer your question: I am (apparantly) not sure what you are asking then.