left coset covered by finite union of right coset Let $H<G$ be a subgroup inclusion with both groups being countable (typically, they are infinite groups). 
I am wondering is there a name for the following property?
$\forall g\in G$, there is a finite subset $K_g\subseteq G$ such that $gH\subseteq HK_g:=\cup_{s\in K_g}Hs$.
Clearly, when $H$ is normal or finite, this property holds, any other interesting examples?
Any reference would be helpful.
 A: The property you describe is equivalent to the property of being a commensurated subgroup.
By definition, $H$ is a commensurated subgroup of $G$ if for all $g \in G$, the index $|H: H \cap gHg^{-1}|$ is finite.
Commensurated subgroups show up in various places in group theory and related areas.  Some cases I am familiar with:
A Hecke pair is just a pair $(G,H)$ where $G$ is a group and $H$ is a commensurated subgroup of $G$.  The terminology comes from the connection with Hecke algebras; I don't know a good single reference, but if you look up 'Hecke pair' you will find various descriptions of algebras associated to such a pair.
You often get commensurated subgroups appearing in the theory of arithmetic groups, for example $\mathrm{SL}_n(\mathbb{Z})$ is commensurated inside $\mathrm{SL}_n(\mathbb{Q})$.  The following article by Shalom and Willis, and the previous articles it refers to, have a lot of information about this phenomenon and what is currently known: https://arxiv.org/abs/0911.1966
Let $G$ be a totally disconnected, locally compact topological group.  Then by Van Dantzig's theorem, $G$ has a base of neighbourhoods of the identity consisting of compact open subgroups.  It's an exercise to check that every compact open subgroup $U$ is commensurated, so $(G,U)$ is a Hecke pair.  The class of totally disconnected locally compact groups is rather diverse, so this is a source of many interesting examples of commensurated subgroups.  Moreover, in some sense one can model all Hecke pairs this way, by means of completing your original group with respect to a uniformity, so the theory of commensurated subgroups is intimately linked with the theory of totally disconnected locally compact groups.  Phillip Wesolek and I have an article about it on the arXiv: https://arxiv.org/abs/1509.00156
Suppose you have an infinite group $H$, such that every non-trivial conjugacy class in $H$ is infinite.  Then you can define the 'abstract commensurator' of $H$, which is the largest group $G$ such that $H$ is a commensurated subgroup of $G$ and such that no element of $G \setminus \{1\}$ centralizes a finite index subgroup of $H$.  (It's like the automorphism group, but with 'normal' replaced by 'commensurated'.)  For more about this idea (in the topological group setting, but you can do the same sort of thing without topology), see this paper of Barnea, Ershov and Weigel: http://www.ams.org/journals/tran/2011-363-10/S0002-9947-2011-05295-5/
For more examples of what can be proved in general about commensurated subgroups, here is a recent article by Pierre-Emmanuel Caprace, Peter Kropholler, Phillip Wesolek and me: https://arxiv.org/abs/1706.06853
