COHOMOLOGY OF 2-GROUPS -- SECOND RUN
THE MOD-2 COHOMOLOGY OF 2-GROUPS
SECOND RUN
On this web page we present the data from the second run of the
computer calculation of the mod-2 cohomology of groups of
order 8, 16, 32 and 64. All of the calculations were made using
the MAGMA computer algebra system. The groups
are indexed by their Hall-Senior Numbers. The first run of the
cohomology rings was terminated in July of 1997. The cohomology
rings of all but five of the groups of order 64 were successfully
calculated. The failure to compute the last five were all due to
problems with the Groebner basis machinery used to make minimal
sets of relations among the generators of the cohomology. To view the
results of the first run
click here.
. The user should be warned at this stage that
the results of the computation on the second run will, in many cases,
not look like the results from the first run. There are several probabilistic
elements in the calculation that cause different
choices of variables for the cohomolgy rings in many of the
calculations. The computed cohomology rings
from the first and second runs will certainly be isomorphic, but direct
comparisons cannot be made.
New Features
In the second run we calculate several new aspects of the cohomolgy
that were not computed in the first. The main items are the following.
Inflation Maps: The images of the inflation maps from all quotient
groups G/H where H had order 2 were computed. In the cases where
the factor group G/H is not abelian, links are provided so that the
user may view the structure of the cohomology ring of G/H.
Nilpotence Indicies of the Nilradical and of the Essential Cohomology:
We find the least integer n such
that In = 0 where I is the nilradical of the cohomolgy
ring -- i. e. the kernel of the restriction to the elementary abelian
2-subgroups of G. We also compute the nilpotency index of the essential
cohomolgy.
Depth-essential cohomology: The depth-essential cohomology is the
intersection of the kernels of the restrictions of the cohomology ring
to the centralizers of the elementary abelian subgroups of rank
2d+1 where d is the computed depth of the
cohomolgy ring. It is an open question as to whether the depth-essential
cohomology must always be nonzero and must have the property
that the dimension H*(G, k)/J is always d and
that the depth-essential cohomology is a finitely generated free module
over the polynomial subring of H*(G, k) generated by a
regular sequence of maximal length. See [Dep] or [Prob] for more
details. Note that the check for the free module property does not
always work, though sometimes it can be done by hand. A warning is
in order here. There was a flaw in the orginal program for this
computations and the result for the groups in the early part of the
calculation may be suspect. This will all be completely corrected
in the third run.
Automorphisms of cohomology: We determine the outer automorphism of
all of the groups that are not abelian and calculate the maps on the
cohomology ring induced by a set of generators for the outer automorphism
group. Some of the programs for this feature were written by Jason
Whitt.
State of the Calculation
The second attempt at the calculations was begun in February of
1998. As of early November (1998)
all but six of the cohomology ring of the 267 groups of
order 64 had been calculated. Some partial results have been
posted for the cohomology rings of the remaining six groups.
As in the first run the problems have been mostly in the
Groebner basis computations. For the third run we are preparing
programs that will circumvent these problems.
The Calculations
Groups of Order 8
Groups of Order 16
Groups of Order 32
Groups of Order 64
Notes and Definitions
Brief explaination of some of the terms and concepts that appear in
the output of the calculations of the mod-p cohomology rings in the
text of the web page for the
first run.
.
Please send me your comments
This web page is very experimental and is not likely to be refereed
or reviewed by anyone except the users. Please feel free to send me any suggestions for
improvement as well as any misprints or mistakes that you might notice.
Equipment
Most of the calculation that are posted were performed on an SUN
ULTRA 2200, (the sloth). The machine has 1024 M. of RAM and approximately
12 G. of hard drive. More notes on the details of the calculations are
given below. I want to thank the National Science Foundation and
University of Georgia Research Foundation for providing me with both
the equipment and the time to work on this project.
Programs
All of the programs are written in MAGMA code and run on the MAGMA
platform. Thanks are due to John Cannon and Allan Steel of the MAGMA
project for numerous instances of help with the tools to make the
programs work and for their enthusiastic support. I am currently working
on adaptations of the programs that will be suitable for inclusion in MAGMA
in the near future.
Some Interesting Features
One of the aims of the project has been to test some of the
conjecture and investigate some of the many questions related to the
structure of group cohomology. Accordingly we point out a few of the
findings of the calculations. These are statements which hold for all
of the cohomology rings that have been calculated. It is likely that
there are many more interesting features that have not been observed
in the limited time that we have had to examine the data.
The maximum of the nilpotency degrees of any of the nilradicals
of the cohomology rings is 7 (group number 108). Almost all of the
others are nilpotent of degree 5 or less. The nilradical of the
cohomology ring of group nunber 79 is nilpotent of degree 6.
The nilpotency degree of the essential cohomology is never more
than 2.
The depth-essential cohomology in nonzero whenever the
cohomology ring is not Cohen-Macaulay.
Moreover the annihilator of the depth-essential cohomology
is an ideal whose variety in the maximal ideal spectrum of the cohomology
ring has dimension equal to the depth of the cohomology ring.
Equivalently, the quotient ring of the cohomology ring by the
annihilator of the depth-essential cohomology has dimension equal
to the depth of cohomology ring.
In the cases that have been calculated the depth-essential
cohomology is a free module over the polynomial subring of the
cohomology ring generated by the elements of a regular sequence of
maximal length.
I would be interested in knowing what other features of the
calculation could be of use or interest for other research in the
area. I plan to make at least one more run through these groups. The
third run should include things like the transfer maps from maximal
subgroup. I am sure that there are many other features worthy of
calculation, and I would welcome suggestions of thing to consider
in the next run.
References
Some of the papers in the bibliography can be obtained
electronically from Dave Benson's electronic archives. Links to
the archive are provided in each case.
[Dep] J. F. Carlson,
Depth
and transfer maps in the cohomology of groups, Math. Z., 218 (1995),
461-468.
[Test] J. F. Carlson,
Calculations of cohomology: Tests for completion, J. Sym. Comp.
{\bf 31}(2001), 229\--242.
[Prob] J. F. Carlson,
Problems in the calculation of group cohomology, Prog. in Math.
{\bf 173}(1999), 107\--120.
[HaSe] M. Hall and J. K. Senior, Groups of order 2n, n
less than or equal to 6, Macmillan (1964), New York
Acknowledgement
Thanks are due to NSF for support of the project in both time and
equipment.