COHOMOLOGY OF 2-GROUPS

# THE MOD-2 COHOMOLOGY OF 2-GROUPS

On this web page we present the data from the first run of the computer calculation of the mod-2 cohomology of groups of order 8, 16, 32 and 64. to see the results of the second run click here.. All of the calculations were run using the MAGMA computer algebra system. The groups are indexed by their Hall-Senior Numbers.

## State of the Calculation

The first run was terminated in July of 1997. The cohomology rings of all but 5 of the 267 groups of order 64 were calculated and are posted.

## The Calculations

Groups of Order 8
Groups of Order 16
Groups of Order 32
Groups of Order 64

## Notes and Definitions

The following document presents a brief explaination of some of the terms and concepts that appear in the output of the calculations of the mod-p cohomology rings. The reader who is interested in more detail is invited to consult the papers [CGS] and [Test]. Other references to specific items are given below. The links in the references are to Dave Benson's electronic archives from which most of the papers can be ftp'd.

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 calculations 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. It is my hope that the programs will be included in MAGMA in some later release. In the meanwhile anyone who is interested may ftp a tar file "cohomology.tar" containing the programs. I apologise for the lack of documentation. I hope to correct that situation in the not too distant future. If you would like a very brief description of what is in the file "cohomology.tar.z" then click here.

## Description of Project

The goal is to compute as many examples as possible of mod-p cohomology rings of finite p-groups. In the early stages we have looked at the groups of order 64, as the groups of order 32 have been done [R]. Ultimately we would like to calculate the cohomology of larger 2-groups, of other p-groups for odd primes p and to use the information to calculate the cohomology rings of finite groups other than p-groups. Some of the last has already been done (e.g. see [CMM]). If there is anyone else who has calculations that they would like to make public then I would be happy to post the files on my server or to establish links to other web pages as appropriate.

The calculations on the sloth run in essentially two stages. The first is all groups, modules and linear algebra, The second stage is all Gröbner basis and commutative algebra calculations. First of all the groups are called out of the library on MAGMA. For each group the MAGMA library number is given though the group is posted according to its HALL-SENIOR number (see [HaSe]). Some routine information about each group is computed and printed. After that the actual calculation of the cohomology begins. For notation let G denote the group and let k be the prime field F2.

### STAGE 1: Modules and Linear Algebra

#### Projective Resolutions

The first step is the computation of a standard free module for the group algebra. In particular, matrices for the action of a minimal set of generators for G and also for a power commutator set of generators for G on the free module kG are fixed. In this way the programs can rely almost entirely on linear algebra. That is, the more time consuming module theoretic machinery is used only at the beginning. Next a minimal projective resolution of the trivial module is computed and stored. Because the rate of growth (complexity) of the projective resolution is related to the 2-rank of G the number of steps that were computed for each resolution varied according to the following formula.

For groups of 2-rank 1, 2 or 3, sixteen steps were computed.
For groups of 2-rank 4, thirteen steps were computed.
For groups of 2-rank 5, ten steps were computed.
For abelian groups only three steps were computed.

Of course, the point about abelian groups is that the cohomology rings are generated in degrees 1 and 2 and the relations are well known. One justification for this scheme is that for groups of large 2-rank, the elements in a system of parameters will lie in low degrees. That is, it can be predicted that if G has order 64 and 2-rank 4 then the elements in a system of parameters will lie in degree 4 or less. If the 2-rank is 5 the parameters will lie in degree 2 or less. This can be proved using norm maps.

#### ChainMaps

Once the projective resolution is obtained the computer begins constructing chain maps for the cohomology generators. This is a careful process and no excess generators are computed. That is, the cup product of any two cohomology elements is obtained as the composition of the corresponding chain maps. The program computes some of the products as it goes and hence the only generators for which the chain maps are calculated are those which are not polynomials in the other generators. The printed output of this stage of the program is a list of the generators and a list of their degrees.

#### Relations

Following the chain map calculation we get the relations among the generators. That is, for a degree n, within the range of the computation, all possible monomials of degree n in the generators are computed. Each such monomial is a k-homomorphism from the nth term, modulo its radical, in the projective resolution into k. Thus the set of relations among the monomials is the null space of the matrix whose rows are the matrices of these homomorphisms. Hence we have determined all possible relations of degree n in the generators. Of course this information is too massive to be helpful. What we really want is a minimal set of generators for the ideal of relations. To get the ideal we need to call upon the Gröbner basis machinery. Hence we are led to Stage 2. The program actually produces a minimal set of generators for the ideal of relations. The notation is taken directly from the output from MAGMA. Hence "*" means multiplication and "^n" means nth power.

### STAGE 2: Commutative Algebra

Once a minimal set of relations has been presented then the computation of the cohomology ring is (probably!) complete. The problem now is to prove it is complete. In [Test] we provide a scheme for the proof. All of this is based on conjectures about the nature of mod-p cohomology rings, but the fortunate thing is that the conjectures can be checked from the output of the computation. From here until the end, all of the calculations are entirely within the domain of the commutative algebra packages on MAGMA.

#### Systems of Parameters and Regular Sequences

The Hilbert series for the cohomology ring and the factorizations of its numerator and denominator are given. This computation required only slight modification of existing algorithms.

A system of parameters for H*(G, k) is a collection of elements, x1, ..., xr with the property they generate a polynomial subring of the cohomology ring for H*(G, k) and moreover the cohomology ring is a finitely generated module over that polynomial subring. Hence it is necessary that r be equal to the 2-rank of G which is the Krull dimension of the cohomology ring of H*(G, k). The program then finds the depth of the cohomology ring by computing the longest regular sequence. There are some problems with the implementation of this routine which are noted below in the section on problems. However the accuracy of the calculation is verified in the next calculation.

#### The hypercohomology Spectral Sequence

The hypercohomology spectral sequence is described in detail in the paper [BC] (see also [Test]). It is a spectral sequence whose E2 page is the Koszul complex of H*(G, k) with respect to the system of parameters chosen above. What is actually printed is the homology of the Koszul complex. In the event that the cohomology ring is Cohen-Macaulay the spectral sequence plays no part in the test for the completion of the calculation and is omitted. In this case the homology would all exist on the bottom row (Row 0). In the event that the depth of the cohomology ring is one less than the Krull dimension then the homology of the Koszul complex lies on two rows. The bottom row (Row 0) is presented by giving bases of the vector spaces of the homogeneous parts of the ring H*(G, k)/x1, ..., xr . The bases for this row and the others are presented by degrees. That is the output

ROW 0:[z,x,w][z*x,u] ...

would indicate that the space in degree one has basis consisting of the elements z, x, and w, the space in degree two has basis consisting of zx and u, etc. The symbol [] is to be interpreted as indicating that the space is the zero space.

#### Restrictions to Elementary Abelian Subgroups

The celebrated theorem of Quillen [Q] tells us that the minimal primes of the cohomology ring H*(G, k) are the radicals of the kernels of restrictions to the conjugacy classes of maximal elementary abelian p-subgroups of G. Because we are only dealing with the case p = 2 the cohomology rings of the elementary abelian p-subgroups are in fact integral domains and hence the minimal primes of the cohomology rings of G are the kernels of restrictions. The restriction map programs are described briefly in [CMM]. A complete list of representatives of the maximal elementary abelian subgroups was computed as part of the early information on the group. In this section is listed the representative, the images of the generators for the cohomology of G as elements in the polynomial ring which is the cohomology of the elementary abelian subgroup, and a list of the generators of the kernel of the restriction map.

The intersection of the minimal primes is the nilradical of the cohomology ring. In the event that this intersection is equal to the ideal of relations of the cohomology ring of G then the output of the program is the statement the the nilradical of the cohomology ring is zero. Otherwise the program prints a list of generators for the nilradical (intersection of the minimal primes) that are not in the ideal of relations for the cohomology ring.

#### Restriction to Maximal Subgroups.

A standard routine in MAGMA computes the maximal subgroups of the group G. Then these subgroups are matched with the isomorphic subgroups of order 32. Note that if the subgroup is abelian then the cohomology of the subgroup is known from the structure of the group. The cohomology ring of the subgroup is then computed out to the maximum of the degrees of the generators of H*(G, k). This is for the purposes of computing the images of the generators of H*(G, k) under the restriction map and also for computing the kernel of the restriction. If the subgroup is not abelian then the cohomology ring of the subgroup is pulled from the record of the cohomology rings of the groups of order 32. Using the constructed isomorphism of the subgroup with the group of the record the restriction map is created and the images of the generators of H*(G, k) in the cohomology ring of the subgroup are determined. Also the kernel of the restriction map is computed.

#### Essential Cohomology

The intersections of the kernels of restriction to the maximal subgroups is known as the essential cohomology. This ideal is constructed using the commutative algebra capabilities in MAGMA. The conjecture in [Test] is that the essential cohomology is nonzero only if the depth of the cohomology ring H*(G, k) is equal to the p-rank of the center of G. Moreover in the case that the essential cohomology is nonzero then it is further conjectured that it is a finitely generated free module over the polynomial subring Q generated by the elements of the longest regular sequence in H*(G, k). The finitely generated part of this last question is easily checked by testing the dimension of the ideal of the essential cohomology using routines which are built into MAGMA. That is, if the dimension of the ideal of the essential cohomology is equal to the p-rank of the center of G, then it is also equal to the length of the longest regular sequence by Duflot's Theorem [Du]. A positive answer to the conjecture would also lend evidence for some of the speculation in [Dep]. A test to show that the essential cohomology is a free module over Q seems to be much more difficult and may be beyond reason at this point. The computer program runs a check on this by seeing if the generators for the essential cohomology over Q are also generators for the cohomology ring H*(G, k) as a module over Q. This guarantees the free module property because it is known from commutative algebra (in the case that p-rank = depth) that the whole cohomology ring is a free module over Q. However the failure of the check does not mean that the conjecture is false. We say more about this below.

### Flags

In the course of the calculation certain flags have been set which record some of the information about the cohomology rings. Some of these are necessary for the proof that the calculation is compete and a complete set of generators and relations for the cohomology ring have been generated. The flags and their meanings are given below. Some of the flags are meant to draw attention to undesirable aspects of the calculation.

#### Paramflag

The paramflag is true if the computed system of parameters restricts to every elementary abelian subgroup as a system of parameters, or at least the cohomology ring of the elementary abelian subgroup is a finitely generated module over the subring generated by the restrictions of the elements in the system of parameters. Hence if the paramflag is true then we are guaranteed that the computed system of parameters is really a system of parameters for the cohomology ring. This check is an essential ingredient in the test for completion.

#### Qregflag

If the qregflag is true then the computed system of parameters is a quasiregular sequence (see [BC] or [Test]) for the computed cohomology ring. That is if, x1, ..., xr is a system of parameters for the computed cohomology ring R then for each i we have that multiplication by xi on R/(x1, ..., xi-1) at least in degrees larger than, deg(x1)+ ...+deg(xr). This check is an essential ingredient in the test for completion.

#### Cmflag

The cmflag is true if the cohomology ring is Cohen-Macaulay. This is not really a part of the test for completion.

#### Essflag

The essflag is true if the essential cohomology is zero. In the case that the essential cohomology is not zero then the essflag is still true provided the the annihilator of the essential cohomology has dimension equal to the p-rank of the center of the group. This would imply that the essential cohomology is a finitely generated module over the polynomial subring Q of the cohomology ring generated by a regular sequence of maximal length. There is one further check that is run in the case that the essential cohomology is nonzero and its annihilator has the right dimension. That is, we check to see if the generators of the essential cohomology as a module over Q are also free generators of the cohomology ring as a module over Q. A failure of this last check does not mean that the essential cohomology is not a free module over Q. But at this point some hand checking is necessary. This check is an essential ingredient in the test for completion.

#### Centflag

The centflag is true is the essential cohomology is zero or if the annihilator of the essential cohomology has dimension larger than the p-rank of the center of the group. If neither of these is true then the depth of the cohomology ring must be equal to the p-rank of the center. So it is important to be certain that the computed longest regular sequence for the cohomology ring is guaranteed to be a regular sequence. The check for this is to make sure that the restriction of these elements to the center of the group is a regular system of parameters. This check is an essential ingredient in the test for completion.

#### Bigflag

In order to keep the computations running the program instructed the machine not to perform certain kernel of restriction programs if the number of variables was too large. In such a case the bigflag was set at false. The negation does not imply that the computation is not complete, but only means that the test for completion has not been run. In the initial posting the computations with false bigflag were left out. We hope that advances in the techniques will eventually take care of all of these problems.

#### Restrictflag

The restrictflag is true as long as the ideal of relations for the cohomology ring is contained in all of the kernels of restriction. This flag is a check on the validity of the program itself and has nothing to do with the calculation.

### Problems

Most of the problems related to the calculations are concerned with the commutative algebra operations. In this section I want to outline some of the difficulties that have been encountered and request help and suggestions from anyone who might know how some of these things should be handled. At some time, I hope to post a collection of challenge problems. These would mostly be problems that applications in MAGMA do not seem to be able to handle. Some of the Gröbner basis calculations may simply be too large to do with the present technology. Others might be possible with application of the right technique. I hope eventually to look at some of the stickier problems with other computer algebra programs. It is a matter of finding the time to do it all.

#### Choosing Variables

The primary problem with the projective resolutions and chain maps is that there is a lot of randomness to the process which sets up the computation. One of the results is that the generators of the cohomology are sometimes not chosen in such a way as to make the relations as simple and compact as possible. It is clear from some of the examples that a simple change of variables would make the ideal of relations easier to understand. Of course, it would be possible to go through the entire set of groups and change the bases one by one. However such an exercise would be extremely tedious and I would much rather figure out a way to have it done automatically. While this may seem like it is only a problem of aesthetics, it may also have some bearing on the Gröbner basis problems.

#### Choosing Systems of Parameters

The current algorithm (of my own writing) for choosing a system of parameters is simply a search in which the machine chooses polynomials, starting with the generators and proceeding up by degrees, and then testing to see if the Krull dimension of the ring drops when the polynomial is factored out. It continues until the Krull dimension becomes zero. For the purposes of the cohomology calculation this is reasonably satisfactory when it works and when it can find a system of parameters quickly. The problems with the technique are two-fold. Sometimes the search is extremely slow, particularly when the parameters cannot be found in low degrees and when there are many generators for the cohomology. This could possibly be cured by extending the field and/or using the theory to predict the degrees in which the parameters will be located. However the dimension test will run more slowly over a larger field. The second problem is that occasionally the output of the calculation is a system of parameters which does not contain a regular sequence of maximal length. When this happens the calculation of the hypercohomology spectral sequence crashes. Again we could try extending the field with the obvious ill effect.

#### Kernels of Restrictions

The cohomology rings of the nonabelian groups of orders 8,16, and 32 are stored in a workspace which is loaded into MAGMA before the computations of the cohomology rings of the groups of order 64 begin. So in particular, it is an easy matter to create a chain map which gives the restriction homomorphism for G to any of its subgroups. Obtaining the images of the generators of the cohomology for G is straightforward and runs in reasonable time. The difficulties start when we try to calculate the kernel of the restriction. This is, of course, an elimination ideal calculation, and such problems are known to be difficult. It has been noted in some examples that improvements in running time can be achieved by changing the ordering on the variables. However, most such improvements are made only on an ad hoc basis and I know of no general formula for dealing with the situation. This problem may turn out to be the most basic stumbling block to getting the cohomology rings of all of the groups of order 64.

## 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.

[BC] D. J. Benson and J. F. Carlson, Projective resolutions and Poincaré duality complexes, Trans. Amer. Math. Soc. 342(1994), 447-488.

[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.

[CGS] J. F. Carlson, E. Green and G. J. A. Schneider Calculating Ext algebras , J. Sym. Comp. 24 (1997), 317-325

[CMM] J, F. Carlson, J. Maginnis and R. J. Milgram, The cohomology of the sporadic groups J2 and J3 , J. Algebra {\bf 214}(1999), 143\--173.

[Du] J. Duflot, Depth and equivariant cohomology, Comment. Math. Helv., 56(1981), 627-637.

[HaSe] M. Hall and J. K. Senior, Groups of order 2n, n less than or equal to 6, Macmillan (1964), New York

[Q] D. Quillen, The spectrum of an equivariant cohomology ring, I, II, Ann. Math., 94(1971), 549-602.

[R] D. Rusin, The cohomology of groups of order 32, Math. Comp., 53(1989), 359-385.

## Acknowledgement

Thanks are due to NSF for support of the project in both time and equipment.