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I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by Alejandro Adem and various other articles by him on calculations of cohomology of finite groups. There is also a book by Adem and Milgram in which they compute cohomology of many finite groups. I would like to know

To what extent the cohomology of finite groups is known. Is it for example known for every finite simple group (in the classification). Any reference on the list of finite groups with known (unknown) cohomology will be nice.

I do apologize if the question is too simple to ask for this forum.

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Two references that naturally come to mind:

  • D.J. Benson and S.D. Smith. Classifying spaces of sporadic groups. AMS Mathematical Surveys and Monographs vol. 147

Describes homotopy colimit decompositions of the $2$-completed classifying spaces of the sporadic groups (based on subgroup complexes from the "$2$-local geometry"). These are relevant for the computation of mod $2$ cohomology. The book also contains remarks on the computations of cup-product structure.

  • J.F. Carlson, L. Townsley, L. Valeri-Elizondo, M. Zhang. Cohomology rings of finite groups. With an appendix: Calculations of cohomology rings of groups of order dividing 64. Algebras and Applications, Volume 3, 2003, Springer.

Describes some of the more advanced methods for computing cohomology rings of finite groups. Looking at the appendix also shows that besides the complications that one would naturally expect with the finite simple groups, computing the cohomology rings of $p$-groups can also be tough business.

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