I am learning algebra and I do not understand the first isomorphism theorem correctly.
I have an injective group homomorphism $\phi: G \to H$. Moreover I have given that $im(\phi)\cong L$, with $L$ a group.
By the first isomorphism theorem it holds: $L \cong G/ker(\phi)$. Since, $\phi$ is injective it holds $ker(\phi)=\{e\}$. Does this mean $G \cong L$?
Best Answer
The assumption ${\rm im}(\phi)\cong L$ says that $\phi \colon G\rightarrow L$ is surjective, the other that it is injective. Hence it is an isomorphism of groups, i.e., $G\cong L$. The first isomorphism theorem should not come to a different conclusion. So you are right.