If $H$ is a subgroup of a group $G$, is there a standard name for subsets of the form $xHy$? Let $H$ be a subgroup of a group $G$. The right (resp. left) cosets of $H$ are the subsets of $G$ of the form $Hg$ (resp. $gH$) for some $g \in G$.
Question. Is there a standard name for the subsets of the form $xHy$ (with $x, y \in G$)?
These sets occur naturally in the study of the power monoid of $G$ (the monoid of all subsets of $G$ under the product $XY = \{xy \mid x \in X, y \in Y\}$). I thought of double cosets or bilateral cosets but it seems to be used for other purpose.
 A: As @user1729 said in comments, these are conjugates of cosets. But they can also be described as left or right cosets of subgroups conjugate to $H$.
So $xHy = xy(y^{-1}Hy)$ is a left coset of the subgroup $y^{-1}Hy$.
And $xHy = (xHx^{-1})xy$ is a right coset of the subgroup $xHx^{-1}$.
I remember a while back someone asked for help with an exercise that said "prove that every left coset in a group is a right coset", and several people replied that this was wrong, but in fact it is correct because $xH = (xHx^{-1})x$.
A: To my knowledge there exists no name for these subsets of $G$. As said in the linked question from the comments, you can identify sets $xHy$ with cosets in $G \times G$ (or $(G \times G) / \{ (g,g) \mid g \in G \}$ as I pointed out in the comments).
However, this action of $G \times G$ on $H$ could be interpreted differently, reminiscent of the action of two rings on a single module, known as a Bimodule. Given $H$, we have a left action and a right action of $G$ on $H$, and both are compatible in the following sense:
$$
 (x^{-1} H)y = x^{-1} (Hy).
$$
This essentially means that, if we interpret the action in the symmetric group over the, for example, right cosets, then the permutations corresponding to the left action commute with the permutations corresponding to right action, i.e, they are contained in the centralizers of each other in the symmetric group.
So, I would suggest to call them bicosets, as was already suggested in the comments of the other question. I never encountered this name before, so, I suppose it is new.
