Recent research in Algebra and Topology

Extracted from the 2001 MSI Annual Report

Algebraic geometry

In geometric invariant theory, Martin (Jerusalem) and Neeman produced a counterexample to a claim made by Bardsley and Richardson about fifteen years ago. Bardsley and Richardson claimed in 2.1.9(b) of their article [Étale slices and transformation groups in characteristic p, Proceedings of the London Mathematical Society, 51 (1985), 295-317] that the map from a variety X to its quotient X/G by a reductive algebraic group is always separable. Martin and Neeman gave a counterexample. In fact, in the counterexample X is a vector space and the G-action is linear.

The relevance of the counterexample is that it can be used to make precise the limits on how far Luna's slice theorem can be extended to characteristic p.

Algebraic K- and L-theory

Neeman and Ranicki (Edinburgh) continued their study of the localisation exact sequence in the K-theory of non-commutative rings. The results, which were still in the formative stage last year, now appear in a preprint that has been submitted for publication. The preprint is part 1 of a series, and Neeman and Ranicki are currently working on part 2.

The results in part 1 say the following. Let R be a ring and S its Cohn-Schofield localisation. Then there are long exact sequences in K- and L-theory as long as Tor_i^R(S,S) vanishes whenever i > 0. The best results up until now were due to Pierre Vogel (Strasbourg), and his results assumed S is flat over R. It is easy to construct examples of S where Tor_i^R(S,S) vanishes, but S is not flat over R. The new results are therefore a genuine improvement over what was known.

Finite groups

A long standing problem in the theory of groups is the following: given an irredundant cover of a group by subgroups, what is the precise bound on the index of the intersection of the cover as a function of the number of subgroups in the cover? Bryce and Serena (Firenze) have outlined a method for reducing the problem to that in which the subgroups of the cover are maximal. They have shown the method to work in a number of small cases.

A number of results in the literature are of the following type: a group is in a given class X of groups if its two-generator subgroups are in X. There are various ways of weakening this requirement. One that has recently appeared in the literature asks whether a group is in X if, whenever M, N are subsets with n elements of the group, there is an element in M and an element in N which together generated a group in X. Bryce has answered this question positively for sufficiently large finite groups and several familiar classes X.

There has been much activity recently in investigating the structure of finite groups that can be written as the product of two subgroups. If the subgroups satisfy extra permutability conditions it is often possible to obtain more detailed information about the structure. Ballester-Bolinches (Valencia), Cossey and Esteban (Valencia) have begun an investigation of the structure of the product of two mutually permutable abelian groups; it is well known that the product is abelian. It appears that in fact the product is almost a modular group, though examples show that the structure is more complex. Cossey and Esteban have begun to extend the results of earlier authors on the product of two finite nilpotent groups of coprime order. The smallest case, for which a precise bound on the derived length of the product in terms of the derived length of the factors is not known, is one factor of derived length 3 and the other abelian. Here they believe the best possible bound is 5 for the odd order case and have constructed an example of derived length 5.

Cossey and Stonehewer (Warwick) have been considering the normal closure of a permutable subgroup of a finite p-group. For a permutable cyclic subgroup of odd order they had shown that the normal closure is abelian. To settle the case of a cyclic permutable subgroup of even order has been much more complicated and has only recently been completed; here the normal closure has class 2. They are now considering the normal closure of an abelian permutable subgroup.

Havas (Qld), O'Brien (Auckland) and Newman have continued their study of minimal presentations for finite groups. Newman worked on generating sets of certain cyclically-presented groups. Eick (Braunschweig), O'Brien (Auckland) and Newman studied space groups with a view to getting further results about the relationship between the breadth and nilpotency class of groups with prime-power order. Wiegold (Cardiff) and Newman have been investigating groups with prime-power order with low breadth.

Ormerod has worked mainly on finite p-groups of Wielandt length 3, aiming at improving some previous results. Finite p-groups of Wielandt length 3, when factored by their Wielandt subgroup, can be nilpotent of class 2 or 3. Previously Ormerod had found characterisations of 3-generator groups when the factor group has class 2, and 2-generator groups when the factor group has class 3. In both cases she considered only the cases where the prime involved is larger than 3. In her current project work she again considers only cases where the prime is larger than 3, and is working towards a characterisation of groups of Wielandt length 3 with any finite number of generators when the factor group by the Wielandt subgroup has class 2. This is nearly complete. She has also done further work on the case when the factor group has class 3, but this is not so complete.

Neumann continued his research into group theory and universal algebras.

Homological algebra

In answer to a question by Marco Schlichting (Urbana), Neeman showed that the derived category of a Grothendieck abelian category need not be compactly generated. In fact, the abelian category can be taken to be the category of sheaves of abelian groups on any positive-dimensional non-compact manifold.

Lie algebras

Bryant (UMIST), Kovács and Stöhr (UMIST) continued to investigate the module structure of free Lie algebras.

Caranti (Trento) and Newman have been making progress on the structure of finite-dimensional graded Lie algebras with maximal class.