Notes:Coset stuff/Quotient group

Problem

I really want like a categorical approach to the quotient group. Not "and look, if we take a normal subgroup then the cosets form a group, how lucky!" the nearest I've got is:

• Applying factoring to:

  [ilmath]\xymatrix{ G\times G \ar@/^3ex/[rr]^{\pi\circ\otimes} \ar[d]_{(\pi,\pi)} \ar[r]^\otimes & G \ar[r]^\pi & \frac{G}{K} \\ \frac{G}{K}\times\frac{G}{K} \ar@{.>}[urr]_{\overline{\otimes}:=\overline{\pi\circ\otimes} } }[/ilmath]

  • Such that [ilmath]\overline{\otimes} [/ilmath] is a group operation on [ilmath]\frac{G}{K} [/ilmath] and [ilmath]\pi[/ilmath] is a surjective group morphism.

But that feels very weak.

It is however without a doubt what we're doing. The normal-groups requirement pops up when trying to show that you can even apply factoring to this case.

[ilmath]\pi:G\rightarrow\frac{G}{K} [/ilmath] is the canonical projection of the equivalence relation, see the parent page (Notes:Coset stuff) for information on that.