Let $G$ be a (finite) group with subgroup $H$. Pick out (left) coset representatives: $x_1=1, x_2, \dots, x_\ell$ (so $x_1H=1H=H$). Now form an $\ell \times \ell$ table with rows and columns labeled by $H, x_2H, \dots, x_\ell H$. 

In addition group cosets together according to their double cosets.
For example, if $Hx_2H=x_2H \cup x_3H \cup x_4H$ we group cosets 2, 3, and 4 together, so we would have something like $(H)$ $(x_2H \; x_3H \; x_4H)$ $(x_5H \cdots$
This (parenthesized) list forms the first row of our table.

The rest of the rows are formed by left multiplying by each representative in turn. So, for example, 
if $Hx_2H=x_2H \cup x_3H \cup x_4H$, we would have

Row $x_1H$: $\qquad \qquad (H) \; (\qquad x_2H \qquad x_3H \qquad x_4H) \; (\qquad x_5H \; \cdots$

Row $x_2H$: $\qquad (x_2H) \; (x_2x_2H \;\; x_2x_3H \;\; x_2x_4H) \;\; (x_2x_5H \; \cdots$

etc.

Notice that in row $xH$, two cosets $yH$ and $zH$ lie in the same parenthesized group if and only if $x^{-1}yH$ and $x^{-1}zH$ lie in the same double coset. Put another way, cosets $yH$ and $zH$ are connected in row $xH$ if and only if $xHx^{-1}yH = xHx^{-1}zH$. So the parentheses in row $xH$ are recording something about the action of $xHx^{-1}$ on the left cosets of $H$. 

Of course if $H$ is a normal subgroup, this is just the Cayley table for the quotient group $G/H$ (with some extra parentheses). 

When $H$ is not normal, switching representatives necessarily changes the order in which cosets appear in rows 2+ but the parenthesized groups stay intact. This pseudo-quotient gets as close to a group structure as one can hope for when $H$ fails to be normal.

My question, does this look familiar to anyone? References?

Now for some much needed explanation. About one year ago I got a call out of the blue by someone living close to my university who said he had a short proof of the Feit-Thompson odd order theorem. 

I was quite skeptical but told him to email me his proof, I'd look it over and we could talk. It turns out he is a retired mathematician and who has worked off-and-on on his proof for 30+ years as a hobby. While I still have serious doubts his "proof" can be adequately formalized and/or corrected, I did find his fundamental object of study interesting -- these "pseudo-quotients" which he call his "diagrams". 

These pseudo-quotients allow one to almost quotient groups even when there are no normal subgroups around. Using these pseudo-quotients, I thought we had a character free proof that the [Frobenius Kernel][1] is a normal subgroup, but it fell apart and I haven't been able to repair the proof.

While these quotients haven't helped me prove anything nontrivial (yet), they have caused me to "trip over" a lot of old important theorems.

  [1]: http://mathoverflow.net/questions/63142/character-free-proof-that-frobenius-kernel-is-a-normal-subgroup