# Sage and group cohomology

I’ve been thinking about computing group cohomology by computer

recenty, I thought I’d point out some features of Sage (the leading

free and open-source alternative to Magma, Maple, Mathematics, etc..).

Specifically, with the 4.1.1 release there was added a package for

computing p-group cohomlogy. Below is the link and description.

Also, if you don’t know about Sage, you can download and install it

here: http://www.sagemath.org/

One nice side-effect of installing Sage is that you get a whole bunch

of other useful mathematics programs pre-compiled, configured, and

installed for you, like GAP (for doing group-theory, symbolic linear algebra, and much more).

http://sage.math.washington.edu/home/SimonKing/Cohomology/

New optional package p_group_cohomology version 1.0.2 (Simon A. King,

David J. Green) #6491 — The package p_group_cohomology can compute the

cohomology ring of a group with coefficients in a finite field of

order p. Its features include:

* Compute the cohomology ring with coefficients in

for any finite p-group, in terms of a minimal generating set and a

minimal set of algebraic relations. We use Benson’s criterion to prove

the completeness of the ring structure.

* Compute depth, dimension, Poincare series and a-invariants of

the cohomology rings.

* Compute the nil radical.

* Construct induced homomorphisms.

* The package includes a list of cohomology rings for all groups

of order 64.

* With the package, the cohomology for all groups of order 128 and

for the Sylow 2-subgroup of the third Conway group (order 1024) was

computed for the first time. The result of these and many other

computations (e.g., all but 6 groups of order 243) is accessible in a

repository on sage.math.