The Cohomology of Finite p-Groups

We present here the cohomology rings of all 2-, 3-, 5- and 7-groups of order at most 128, of all but 6 groups of order 243, and of some sporadic examples of order up to 1024.

For each cohomology ring, we provide dimension and depth, the Poincaré series, the a-invariants, a minimal generating set and minimal algebraic relations between the generators.

We also provide some background information on our computational approach.

Our work meanwhile resulted in a Sage package (Documentation and Installation)

2-Groups 3-Groups 5-Groups 7-Groups
Order 2
Order 4
Order 8
Order 16
Order 32
Order 64
Order 128
Order 256
Order 512
Order 1024
Order 3
Order 9
Order 27
Order 81
Order 243
Order 729
Order 5
Order 25
Order 125
Order 625
Order 7
Order 49
Order 343

Acknowledgment

The work that we are presenting here was supported by the German Science Foundation (DFG), project number GR 1585/4–1.

Simon King
School of Mathematics, Statistics & Applied Mathematics
National University of Ireland, Galway
University Road
Galway (Note that there is no ZIP code in Ireland)
Ireland
E-mail: simon dot king at nuigalway dot ie
Tel: +353 (0)91 49 3175
Fax: +353 (0)91 49 4542
Office: C212, Áras de Brún

Last change: 07/31/2009