Set of Double Cosets
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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.
group-theory group-actions
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