Universal property of quotient group

Question

Let $H\le G$ be a normal subgroup of $G$. Then we can think a natural projection map $\pi:G \to G/H$.

This map has the following universal property:

Let $\phi:G \to G'$ be a homomorphism. If $H \subset \ker (\phi)$, there is a unique homomorphism $\tilde{\phi}:G/H \to G'$ such that $\phi=\tilde{\phi} \circ \pi$.

I think it should be the universal property of the natural projection map. However, if the natural projection map $q$ is eligible to be called to have the universal property, then such map $\pi$ should be unique. That is,

Let $f:G\to K$ be a surjective homomorphism which has the following property: For any homomorphism $\phi:G \to G'$ satisfying $H \subset \ker (\phi)$, there exists a unique homomorphsim $\tilde{\phi}:K\to G'$ such $\phi=\tilde{\phi} \circ f$, then $\ker (f)=H$.

But I can't prove it. Would you let me know how to prove it? Or if it is not true, how can we modify the above universal property of natural projection map?

Any comment will be appreciated!

Answer 1

I think the statement as is not quite correct; there is a easy fix however, but first take a look at the following counterexample:

Let $G = \mathbb Z,\ K = \mathbb Z/4$ and $H = 2\mathbb Z$. The projection map $f: \mathbb Z \to \mathbb Z/4$ has the property that for any map $\phi: \mathbb Z \to G'$ such that $2\mathbb Z\subset \ker(\phi)$ (which implies $4\mathbb Z \subset \ker(\phi)$), there is a unique map $\tilde\phi:\mathbb Z/4\to G'$ such that $\phi = \tilde\phi \circ f$. However, clearly the kernel of the map $\mathbb Z\to \mathbb Z/4$ is $4\mathbb Z$, not $H$.

The fix is to require that the map $f:G\to K$ has kernel containing $H$, i.e. $H\subset \ker(f)$. In that case, letting $G' = G/H$ in your statement, we get a unique map $K\to G/H$, and letting $G' = K$ in the universal property of quotient maps, we get a unique map $G/H\to K$. Composition of these two maps $K\to G/H$ and $G/H\to K$ must be identity because of the universal property in your statement. We conclude that $K$ and $G/H$ are isomorphic, and hence $\ker(G\to K) = H$, as desired.

Remark: the proof just given is very typical of the principle "universal objects are unique upto unique isomorphism" in category theory. If you are curious, I recommend skimming the first few pages of Ch.1 of Ravi Vakil's Foundations of Algebraic Geometry (http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf)