Consider a group $G$ with subgroups $H$ and $K$. The double cosets $\{ HgK : g \in G \}$ partition $G$. It seems this is just a set, not a group action. But I thought I might be missing something- can this be viewed as a group action of some group? I suppose it is the set of $K$-orbits of the group action of $G$ on $G/H$ (and vice versa). Also, I thought perhaps that $G$ could act, where for $g, g_0 \in G$, $g \cdot Hg_0 K = H g g_0 K$, but it seems this is not well defined.
More generally, I would appreciate any answer stating whether there is a universal property of this construction in some category.
Best Answer
If I understand your first question correctly, then the answer is that these double cosets are the orbits of the (right) action of the direct product $H \times K$ on $G$ given by $$ g^{(h, k)} = h^{-1} g k. $$ (I denote the action by an exponent.)