We study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? In this paper we provide, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens–Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.

We develop a theory of 2-fusion systems of even characteristic, and use that theory to show that all S3-free saturated 2-fusion systems are constrained. This supplies a new proof of Glauberman's Theorem on S4-free groups and its various corollaries.

Let F be an algebraically closed field, G be a finite group and H be a subgroup of G. We answer several questions about the centralizer algebra FGH. Among these, we provide examples to show that

• • the centre Z(FGH) can be larger than the F-algebra generated by Z(FG) and Z(FH),

• • FGH can have primitive central idempotents that are not of the form ef, where e and f are primitive central idempotents of FG and FH respectively,

• • it is not always true that the simple FGH-modules are the same as the non-zero FGH-modules HomFH(S, T ↓ H), where S and T are simple FH and FG-modules, respectively.

When a minimal Heilbronn character θ is unfaithful on a Sylow p-subgroup P of a finite group G, we know that G is quasi-simple, p is odd, P is cyclic, NG(P) is maximal and either NG(P) is the unique maximal subgroup containing Ω1(P) or G/Z(G) ≅ L2(q) for q an odd prime with p dividing q − 1. In this paper we examine the exceptional case, where G/Z(G) ≅ L2(q), explicitly constructing unfaithful minimal Heilbronn characters from the non-principal irreducible characters of G.

Many problems about local analysis in a finite group G reduce to a special case in which G has a large normal p-subgroup satisfying several restrictions. In 1983, R. Niles and G. Glauberman showed that every finite p-group S of nilpotence class at least 4 must have two characteristic subgroups S1 and S2 such that, whenever S is a Sylow p-subgroup of a group G as above, S1 or S2 is normal in G. In this paper, we prove a similar theorem with a more explicit choice of S1 and S2.

We define sparse saturated fusion systems and show that, for odd primes, sparse systems are constrained. This simplifies the proof of the Glauberman–Thompson p-Nilpotency Theorem for fusion systems and a related theorem of Stellmacher. We then define a more restrictive class of saturated fusion systems, called extremely sparse systems, that are constrained for all primes.

We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds's recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p-groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p-group P.

Let Un(q) denote the upper triangular group of degree n over the finite field with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form Uk(q) ∩ wUk(q), where w is a monomial matrix in GLk(q) for some k < n.

It follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories , being a Mackey functor on is equivalent to being extendible to a suitable inverse category containing . We further show that extensions of inverse categories by abelian groups are again inverse categories.

Bob Oliver conjectures that if p is an odd prime and S is a finite p-group, then the Oliver subgroup contains the Thompson subgroup Je(S). A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using the ideas and work of Glauberman, we prove that if p ≥ 5, G is a finite p-group, and V is an elementary abelian p-group which is an F-module for G, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in G. We apply this to show that Oliver's Conjecture holds provided that the quotient has class at most log2(p − 2) + 1, or p ≥ 5 and G is equal to its own Baumann subgroup.

After summarizing from previous papers the definitions of the concepts associated with nets, i.e. triples of 6-transpositions in the Monster up to braiding, we give some results.

We find conditions which imply that a saturated fusion system over a product of 2-groups splits as a product of fusion systems over the factors.

For any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

Consider the Mackey functor that assigns to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p > 0. Thévenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source.

Let be a finite category and let k be a field. We consider the category algebra and show that -mod is closed symmetric monoidal. Through comparing with a co-commutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. In particular, we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove that the Hochschild cohomology rings of a certain type of finite category algebras are finitely generated.