can I conclude that image of a function is bijective to its domain by proving its injective?

Question

I have a homomorphism (function) $\phi: G_1 \longrightarrow G_2$ and I have showed that $\phi$ is injective (I showed that kernel($\phi$) = identity of $G_1$)

can I conclude that $G_1 \cong im(\phi)$? in other words, is $G_1$ isomorphic to the image of $\phi$? I think the definition of a "image" implies the onto property?

Answer 1

Yes, $G$ modulo the kernel is isomorphic to the image. If the kernel is trivial then $G$ is isomorphic to the image. I assume we are talking about groups here.

Answer 2

Yes. Surjectivity is not as intrinsic as injectivity is. It depends on the choice of the codomain. As you say, you can make any function surjective by changing the codomain to be the image.