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Sage and group cohomology

September 22, 2009

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

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).


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 \mathbf{F}_p
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.


From → computers, math

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