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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 $\{xHy\,|\,x,y\in G\}?$

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$xHy=((yx)H)^y$ so its a conjugate of a left coset. I'm not sure if there's a term or not. –  Alexander Gruber Mar 13 '13 at 18:19
    
Maybe one would call it a bicoset? I doubt there's any standard term, though. –  Avi Steiner Mar 13 '13 at 18:20
    
What would you use such a set for? –  Qiaochu Yuan Mar 13 '13 at 18:58
    
@QiaochuYuan They came up in my attempt at understanding how products of subsets of $G$ work. I'm not sure yet what I want to do with them because it's all a bit blurry, but they seem significant at the moment. I thought if I had a name I could read up. –  Bartek Mar 13 '13 at 19:02
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It appears that whatever you call them, it should not be double coset (although it sounds like a sensible name) because this already refers to sets of the form $HxK$ for two subgroups $H,K$ and an element $x$. –  Matt Pressland Mar 13 '13 at 22:10
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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 $(H \times H)(hx^{-1}, ky) \mapsto x h^{-1} H ky = xHy$.

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