Abelian extensions of topological vector groups

The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group $\text{G}\text{F}$ are topological vector groups is investigated. It is shown that, if F is a real Fréchet group and $\text{G}\text{F}$ a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if $\text{G}\text{F}$ is finite dimensional or if F is one dimensional and $\text{G}\text{F}$ a separable Hilbert group.     

Derived length and conjugacy class sizes

Let G be a finite solvable group, and let F(G) be its Fitting subgroup. We prove that there is a universal bound for the derived length of G/F(G) in terms of the number of distinct conjugacy class sizes of G. This result is asymptotically best possible. It is based on the following result on orbit sizes in finite linear group actions: If G is a finite solvable group and V a finite faithful irreducible G-module of characteristic r, then there is a universal logarithmic bound for the derived length of G in terms of the number of distinct r^′-parts of the orbit sizes of G on V. This is a refinement of the author's previous work on orbit sizes.     

Identities and isomorphisms of graded simple algebras

Let G be an arbitrary abelian group and let A and B be two finite dimensional G-graded simple algebras over an algebraically closed field F such that the orders of all finite subgroups of G are invertible in F. We prove that A and B are isomorphic if and only if they satisfy the same G-graded identities. We also describe all isomorphism classes of finite dimensional G-graded simple algebras.     

Positive laws in fixed points of automorphisms of finite groups

Suppose that a Frobenius group FH with cyclic kernel F and complement H acts on a finite group G in such a manner that C_G(F)=1 and C_G(H) satisfies a positive law of degree k. It is shown that G satisfies a positive law of degree that is bounded solely in terms of k and |FH|.     

Automorphic Plancherel density theorem

Let F be a totally real field, G a connected reductive group over F, and S a finite set of finite places of F. Assume that G(F ?_? ?) has a discrete series representation. Building upon work of Sauvageot, Serre, Conrey-Duke-Farmer and others, we prove that the S-components of cuspidal automorphic representations of $G\left( {\mathbb{A}_F } \right)$ are equidistributed with respect to the Plancherel measure on the unitary dual of G(F _S ) in an appropriate sense. A few applications are given, such as the limit multiplicity formula for local representations in the global cuspidal spectrum and a quite flexible existence theorem for cuspidal automorphic representations with prescribed local properties. When F is not a totally real field or G(F ?_? ?) has no discrete series, we present a weaker version of the above results.     

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.     

Smooth actions of finite Oliver groups on spheres

In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S, and which real G-vector bundles ν over F occur as the equivariant normal bundles of F in S? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G, F, and ν. We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Oliver's equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.     

Infinite graphs with the least limiting eigenvalue greater than -2

Let λ(F) be the least eigenvalue of a finite graph F. The least limiting eigenvalue λ(G) of a connected infinite graph G is defined by λ(G)=inf_F{λ(F)}, where F runs over all finite induced subgraphs of G. In [4] and [5] it is proved that λ(G)⩾−2 if and only if G is a generalized line graph. In this paper all connected infinite graphs (thus all generalized line graphs) with λ(G)>−2 are characterized.     

Primitive words,free factors and measure preservation

Let F _k be the free group on k generators. A word w ∈ F _k is called primitive if it belongs to some basis of F _k . We investigate two criteria for primitivity, and consider more generally subgroups of F _k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H ≤ J ≤ F _k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F _k is primitive. Again let w ∈ F _k and consider the word map w: G × … × G → G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G × … × G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation, and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and, in particular, prove the conjecture for k = 2. It was asked whether the primitive elements of F _k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F ₂.     

Free lattice-ordered groups and the space of left orderings

For a left-orderable group G, let LO(G) denote its space of left orderings, and F(G) the free lattice-ordered group over G. This paper establishes a connection between the topology of LO(G) and the group F(G). The main result is a correspondence between the kernels of certain maps in F(G), and the closures of orbits in LO(G) under the natural G-action. The proof of this correspondence is motivated by earlier work of McCleary, which essentially shows that isolated points in LO(G) correspond to basic elements in F(G). As an application, we will study this new correspondence between kernels and the closures of orbits to show that LO(G) is either finite or uncountable. We will also show that LO(F _n ) is homeomorphic to the Cantor set, where F _n is the free group on n?>?1 generators.     

When are all group codes of a noncommutative group Abelian (a computational approach)?

Let G be a finite group and F be a field. Any linear code over F that is permutation equivalent to some code defined by an ideal of the group ring FG will be called a G-code. The theory of these ??abstract?? group codes was developed in 2009. A code is called Abelian if it is an A-code for some Abelian group A. Some conditions were given that all G-codes for some group G are Abelian but no examples of non-Abelian group codes were known at that time. We use a computer algebra system GAP to show that all G-codes over any field are Abelian if |G|?<?128 and |G| ? {24, 48, 54, 60, 64, 72, 96, 108, 120}, but for F?=? $ {\mathbb{F}_5} $ and G?=?S₄ there exist non-Abelian G-codes over F. It is also shown that the existence of left non-Abelian group codes for a given group depends in general on the field of coefficients, while for (two-sided) group codes the corresponding question remains open.     

On exceptional pq-groups

A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G,in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F,the zeta function of F does not appear.In the present paper we describe effectively all exceptional groups of orders divisible by exactly two prime numbers p and q,which have unique subgroups of orders p and q.     

F-Planar graphs

An F-planar graph, where F is an ordered field, is a graph that can be represented in the plane F × F, with non-crossing line segments as edges. It is shown that the graph G is F-planar for some F if and only if every finite subgraph of G is planar.     

A recognition algorithm for the intersection graphs of paths in trees

Let F be a finite family of non-empty sets. An undirected graph G is an intersection graph for F if there is a one-to-one correspondence between the vertices of G and the sets of F such that two sets have a non-empty intersection exactly when the corresponding vertices are adjacent in G. If this is the case then F is said to be an intersection model for the graph G. If F is a family of paths within a tree T, then G is called a path graph. This paper proves a characterization for the path graphs and then gives a polynomial time algorithm for their recognition. If G is a path graph the algorithm constructs a path intersection model for G.     

The bounded Lie Engel property on torsion group algebras

Let F be a field of characteristic different from 2 and G a group with involution ∗. Extend the involution to the group ring FG, and write (FG)⁻ for the Lie subalgebra of FG consisting of the skew elements. We classify the torsion groups G having no elements of order 2 such that (FG)⁻ is bounded Lie Engel.     

F-Sets in graphs

A subset S of the vertex set of a graph G is called an F-set if every α?Γ(G), the automorphism group of G, is completely specified by specifying the images under α of all the points of S, and S has a minimum number of points. The number of points, k(G), in an F-set is an invariant of G, whose properties are studied in this paper. For a finite group Γ we define k(Γ) = max{k(G) | Γ(G) = Γ}. Graphs with a given Abelian group and given k-value (k ≤ k(Γ)) have been constructed. Graphs with a given group and k-value 1 are constructed which give simple proofs to the theorems of Frucht and Bouwer on the existence of graphs with given abstract/permutation groups.     

E-constrained groups

In this paper we characterize the finite groups G with F*(G) = E(G).     

Lie properties of symmetric elements in group rings II

Let F be a field of characteristic different from 2, and G a group with involution ∗. Write (FG)⁺ for the set of elements in the group ring FG that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if G has no 2-elements, and (FG)⁺ is Lie nilpotent (resp. Lie n-Engel), then FG is Lie nilpotent (resp. Lie m-Engel, for some m). Here, we classify the groups containing 2-elements such that (FG)⁺ is Lie nilpotent or Lie n-Engel.