Let $(G,\ast)$, $(H,\cdot)$ be groups and let $f:G\to N$ be a group homomorphism with the kernel $\ker(f).$ Then $(G/\ker(f),\circ)$ is a factor group $(G/\ker(f),\circ)$ and the function $$\phi:G/\ker(f)\to \operatorname{im}(f),\quad \phi(a\ker(f))=f(a)$$ between this factor group and the group of the image $\operatorname{im}(f)$ is an isomorphism, i.e. both groups are isomorphic.
Proofs: 1