TITLE: If $H$ is a subgroup of $G$ and $x,y\in G$, what is $xHy$ called?
QUESTION [8 upvotes]: For a group $G$, its subgroup $H$ and $x,y\in G,$ we call $xH$ a left coset of $H,$ and we call $Hy$ a right coset of $H.$ Is there a special name for sets of the form $xHy$? Is there a name or notation for the family of sets $\{xHy\,|\,x,y\in G\}?$

REPLY [2 votes]: Alexander and Serkan point out that the individual elements are just special types of cosets: $xHy = (xy) H^y = H^{x^{-1}} (xy)$. This is a special subset of $G$ that is both a left and right coset (of possibly different but conjugate subgroups).
I don't believe it has a special name.
A different idea that is better suited to the entire collection $\{ xHy : x,y \in G \}$ is that $G \times G$ acts on $\{ xHy : x,y \in G \}$ via $(xHy)^{(r,s)} = r^{-1} x H ys$. The action is clearly transitive, and the stabilizer of $H$ is $H \times H$. Hence I would describe it as the space of cosets $(G\times G)/(H \times H)$. The correspondance between the usual definition is $(H \times H)(x^{-1},y) \mapsto xHy$. This is well defined because if $h,k \in H$ then $(H \times H)(hx^{-1}, ky) \mapsto x h^{-1} H ky = xHy$.