# D8 revisited

I have previously calculated the A:sub:`∞-structure for the
cohomology ring of
D_{8} <http://blog.mikael.johanssons.org/archive/2006/11/an-a-structure-on-the-cohomology-of-d8/>`__.
Now, while trying to figure out how to make my work continue from here,
I tried working out what algebra this would have come from, assuming
that I can adapt Keller's higher multiplication theorem to group
algebras.

A success here would be very good news indeed, since for one it would
indicate that such an adaptation should be possible, and for another it
would possibly give me a way to lend strength both to the previous
calculation and to a conjecture I have in the calculation of group
cohomology with A_{∞} means.

So, we start. We recover, from the previous post, the structure of the
cohomology ring as *k[x,y,z]/(xy)*, with *x,y* in degree 1, and *z* in
degree 2. Furthermore, we have a higher operation, *m:sub:`4`*, with
*m:sub:`4`(x,y,x,y)=m:sub:`4`(y,x,y,x)=z*.

*a,b*the generators, dual to

*x,y*, we end up with the maps

As such, this is an unqualified success. We recover our original group
algebra presentation from the A_{∞}-structure, and thus should be
able to do similarily as a test for completion of future calculations as
well. This, of course, needs to be proven before relied upon, but it
lends credence to my hopes.