The group of endo-permutation modules

Let B be the Burnside ring considered as a globally-defined Mackey functor, and k be a field of positive characteristic q. We prove that, when k ⊗ℤ B is restricted to nilpotent groups of order prime to q, the only simple Mackey functors that can appear as subquotients of it are of the form SE,k, where E is the direct product of elementary abelian p-groups. In the special case k = 𝔽2, the simple Mackey functor SE,𝔽2 must appear as a filtration factor in 𝔽2 ⊗ℤ B for every direct product of elementary abelian p-groups E of odd order.

This paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets and of Galois morphisms of G-sets, respectively. The well-known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural Green biset functor structure. The functorial structure of unit groups of these rings is also discussed.

Diagonal p-permutation functors

The Burnside ring A{G) of an arbitrary compact group G is defined in analogy to torn Dieck's definition for a compact Lie group.A topology is introduced on the set of conjugacy classes of closed subgroups, and a theorem of Dress on the topology of the prime spectrum of A(G) for profinite G is improved and extended to this general setting.This paper is condensed from part of my doctoral thesis [5], written under the supervision of Professor T. torn Dieck, to whom I am indebted for encouragement and advice.1.The Burnside ring and its additive structure.Suppose G is a compact group, and define a G-manifold as a differentiable manifold M together with a continuous left G-action such that, for each g G G, the translation g : M - M is a diffeomorphism.Examples are real G-representations and manifolds invariantly imbedded in such.Theorem 1.1.If G is a compact group, M a compact effective G-manifold, then G is a Lie group and M is a differentiable G-manifold.Proof.Since M is compact and locally connected, it has finitely many connected components, say M = Ji"A/,.For 1 < / < n, set G, = {g G GI g(M) = A/,}.This is a subgroup of G, and, since for any w, G M, the map /: G- M via f(g) = g-m is continuous, while G, = f~x(M), G, is clopen in G. Thus H = n"G, is a clopen normal subgroup.Now set H = {h E H\im G M, h-m = m], so that H is a normal subgroup of H and H/H acts effectively on A/,.Then since M is connected, Montgomery-Zippin [7, p. 208, Theorem 2] shows /////, to be a Lie group.Hence H/C\"xHi is again a Lie group.But since G operates effectively.Pi?//, = {h G //|Vw G M, h m = m) = {e}, so that H is a Lie group, and with it G since it is open in G.That M is then a differentiable G-manifold follows from [7, p. 208, Theorem 3].Now for any G-space X, let t(X) = {g G G|Vx G X,g x = x) be that part of G which acts trivially.Then for a G-manifold M, the G-action is derived from an effective G/i(A/)-action via the projection G-G/i(A/).Hence G/t(M) is a Lie group.

Classical algebraic K-theory: The functors K0, K1, K2

The Burnside ring of fusion systems

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

Let A be an abelian group and let K be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in A. In this thesis we prove foundational properties of the A-fibered Burnside ring functor B^A_K as an A-fibered biset functor over K. This includes the determination of the lattice of subfunctors of B^A_K and the determination of the composition factors of B^A_K. The results of the paper extend results of Coşkun and the author for the A-fibered Burnside ring functor restricted to p-groups and results of Bouc in the case that A is trivial, i.e., the case of the Burnside ring functor over fields of characteristic zero.

We prove that an abelian group can support a directed, interpolation order if and only if it is torsion-free or its quotient by its torsion subgroup is noncyclic. The proof is of an elementary nature. As a consequence of the proof, it is also shown that an abelian group can support a directed, interpolation order if and only if it can support a directed, interpolation, weakly semi-isolated order. The paper is completely self-contained so as to be readable by nonspecialists.

Poor modules with no proper poor direct summands

This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

Normalizing isomorphisms between Burnside rings

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic orientations that are known to be counted in the same way. A transversal of the set of equivalence classes is given.

We consider various generalizations of the effective exponentiation problem, also known as the addition chains problem in additive terminology. We attempt to evaluate and consistently compare the results and the methods that were used to obtain them. The considered generalizations are complexity problems (i.e. finding the minimum number of required operations) for the computation a monomial of many variables, a set of powers, elements of finite Abelian groups, elements of free semigroups, systems of monomials, systems of integer linear forms, systems of elements of free Abelian groups along with some other problems. For all these directions of research we overview both the classical results and the results of the author and his followers.

A blackbox secret sharing (BBSS) scheme works in exactly the same way for all finite Abelian groups G; it can be instantiated for any such group G and only black-box access to its group operations and to random group elements is required. A secret is a single group element and each of the n players’ shares is a vector of such elements. Share-computation and secret-reconstruction is by integer linear combinations. These do not depend on G, and neither do the privacy and reconstruction parameters t, r. This classical, fundamental primitive was introduced by Desmedt and Frankel (CRYPTO 1989) in their context of “threshold cryptography.” The expansion factor is the total number of group elements in a full sharing divided by n. For threshold BBSS with t-privacy (\(1\le t \le n-1\)), \(t+1\)-reconstruction and arbitrary n, constructions with minimal expansion \(O(\log n)\) exist (CRYPTO 2002, 2005).