Group rings with metabelian unit groups

Let F be a field of odd characteristic and G a group. In 1991 Shalev established necessary and sufficient conditions so that the unit group of the group ring FG is metabelian when G is finite. Here, in the modular case, we do the same without restrictions on G. In particular, new cases emerge when G contains elements of infinite order.     

Commutative Moufang loops and Bessel functions

LetF be a field. For eachk>1, letG be a finite group containing{x ₁,...,x _k }!×{y ₁,...,y _k}!. Then in the group algebraFG, $$co\dim _F \sum\limits_{j = 1}^{k - 1} {(1 + (x_j x_{j + 1} ))(1 + (y_j y_{j + 1} ))FG = \frac{{|G|}}{{2\pi i}}\oint\limits_{|z| = 1} {\frac{{dz}}{{J_0 (2\sqrt z )z^{k + 1} }}.} } $$ Connections with the theory of commutative Moufang loops are discussed, including a conjectured answer to Manin's problem of specifying the 3-rank of a finitely generated free commutative Moufang loop.     

Exterior algebra structure on relative invariants of reflection groups

Let G be a reflection group acting on a vector space V (over a field with zero characteristic). We denote by S(V ^*) the coordinate ring of V, by M a finite dimensional G-module and by ?? a one-dimensional character of G. In this article, we define an algebra structure on the isotypic component associated to ?? of the algebra ${S(V^*) \otimes \Lambda(M^*)}$ . This structure is then used to obtain various generalizations of usual criterions on regularity of integers.     

Self-duality in the class of precompact groups

A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ:G→G^∧ of G onto the dual group G^∧ (such that Φ(x)(y)=Φ(y)(x) for all x,y∈G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κω=κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.     

The tame infinitesimal groups of odd characteristic

Given an infinitesimal group G over an algebraically closed field k of characteristic p?3, we provide criteria for the principal block B₀(G) of its algebra of distributions to be of tame representation type. These are employed in conjunction with Galois coverings to determine the structure of G modulo its multiplicative center as well as the quiver and the relations of the algebra B₀(G).     

Large element orders and the characteristic of finite simple symplectic groups

The characteristic of a simple group of Lie type is the characteristic of the field over which this group is defined. Let G = Sp_2n (q), where q = 2 ^k . It is shown that every finite group of Lie type with the same two largest element orders as G has characteristic 2.     

Distance matrix polynomials of trees

Let G be a finite connected graph. If x and y are vertices of G, one may define a distance function d_G on G by letting d_G(x, y) be the minimal length of any path between x and y in G (with d_G(x, x) = 0). Thus, for example, d_G(x, y) = 1 if and only if {x, y} is an edge of G. Furthermore, we define the distance matrix D(G) for G to be the square matrix with rows and columns indexed by the vertex set of G which has d_G(x, y) as its (x, y) entry. In this paper we are concerned with properties of D(G) for the case in which G is a tree (i.e., G is acyclic). In particular, we precisely determine the coefficients of the characteristic polynomial of D(G). This determination is made by deriving surprisingly simple expressions for these coefficients as certain fixed linear combinations of the numbers of various subgraphs of G.     

A duality between locally compact groups and certain Banach algebras

We make precise the following statements: B(G), the Fourier-Stieltjes algebra of locally compact group G, is a dual of G and vice versa. Similarly, A(G), the Fourier algebra of G, is a dual of G and vice versa. We define an abstract Fourier (respectively, Fourier-Stieltjes) algebra; we define the dual group of such a Fourier (respectively, Fourier-Stieltjes) algebra; and we prove the analog of the Pontriagin duality theorem in this context. The key idea in the proof is the characterization of translations of B(G) as precisely those isometric automorphisms Φ of B(G) which satisfy ∥ p ? e^iθΦp ∥² + ∥ p + e^iθΦp ∥² = 4 for all θ ∈ $\text{R}$ and all pure positive definite functions p with norm one. One particularly interesting technical result appears, namely, given x₁, x₂?G, neither of which is the identity e of G, then there exists a continuous, irreducible unitary representation π of G (which may be chosen from the reduced dual of G) such that π(x₁) ≠ π(e) and π(x₂) ≠ π(e). We also note that the group of isometric automorphisms of B(G) (or A(G)) contains as a (“large”) ^.closed, normal subgroup the topological version of Burnside's “holomorph of G.”