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A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low degrees, and I'd like to determine where to search for preexisting computations.

Specifically, I am interested in:

  • Finite groups such as $D_{2n}$; $A_n$ and $S_n$ for $n\le 5$; binary dihedral/tetrahedral/etc. groups. In general, small and/or well-studied groups.
  • The ring structures for their group cohomology with $\mathbb Z$ and $\mathbb Z/2$ coefficients, at least in degrees up to about 6.

I don't mind computing these myself, but if they've already been computed it would be nice to know where to look, and these are the kinds of groups whose cohomology rings have presumably already been computed. But when I search for such computations, I can usually find cohomology groups, but not the ring structure. I suspect I'm looking in the wrong places.

So, what are some good starting places to search for preexisting computations of the ring structure on group cohomology?

Examples of what I might hope for:

  • It would be wonderful if there were a database with this information, but this is a lot to hope for.
  • I've had some luck finding computations in papers which use group cohomology on the way to some other result (e.g. computing bordism groups of $BG$), but there are surely plenty of applications I'm unaware of, so if you know of applications that require these kinds of computations as input, I'd be interested in hearing about them.
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    $\begingroup$ For $\mathbb Z$ coefficients, it is a bit difficult. But for ${\mathbb Z}/2$ coefficients, for $S_4$, you can find in Adem & Milgram, for $D_n$, probably it is also in Adm-Milgram, but it is certainly in MacLane. For $S_5$ and $A_5$, it follows from the computation of $S_4$ and $A_4$ if you only care about mod $2$ coefficient, but it is unlikely to find an "explicit" expression for these. Well, probably you can look Kechagias for cohomology ring of $S_n$ in general. Anyway, the ring structure, in these case, is known completely (for mod 2 coefficients). $\endgroup$ – user43326 Nov 21 at 19:53
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    $\begingroup$ For dihedral groups: Handel's paper "On products in the cohomology of the Dihedral groups". $\endgroup$ – Chris Gerig Nov 21 at 20:36
  • $\begingroup$ @user43326 and Chris Gerig: thank you for the references! $\endgroup$ – Arun Debray Nov 22 at 15:37
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    $\begingroup$ I would also recommend King's site. But since $S_n$ in general was mentioned, let me cite front.math.ucdavis.edu/0909.3292 with mod-p done by Guerra (published at AGT) and and for $A_n$ mod-two front.math.ucdavis.edu/1705.01141 $\endgroup$ – Dev Sinha Nov 26 at 20:31
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Simon King and David Green maintain a computer calculated computation of the mod p cohmology of many finite $p$-groups ('order at most 128, of all but 6 groups of order 243, and of some sporadic examples of order up to 1024') here.

Simon King also maintains computations of the mod p cohomology of certain other finite groups here.

There is also an excellent book 'Cohomology Rings of Finite Groups' by Carlson, Townsley, Valero-Elizondo and Zhang.

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  • $\begingroup$ That's the answer I would have given too. $\endgroup$ – Nicholas Kuhn Nov 22 at 2:54
  • $\begingroup$ This looks like exactly what I was hoping for. Thank you! $\endgroup$ – Arun Debray Nov 22 at 15:38

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