Pseudofinite homogeneity, isolation, and reducibility

It has been proved (by S. M. Dudakov and M. A. Taitslin) that the reducibility of some models of a theory implies the second pseudofinite homogeneity property for this theory. We prove the converse, namely, that any theory with the first or the second pseudofinite homogeneity property has a reducible model and, therefore, possesses the second isolation property. This also proves the equivalence of the second isolation property and the second pseudofinite homogeneity property, in contrast to the first pseudofinite homogeneity property, which is more general than the first isolation property (this was established by O. V. Belegradek, A. P. Stolboushin, and M. A. Taitslin).     

Graphical Frobenius representations

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A graphical Frobenius representation (GFR) of a Frobenius group G is a graph whose automorphism group, as a group of permutations of the vertex set, is isomorphic to G. The problem of classifying which Frobenius groups admit a GFR is a natural extension of the classification of groups that have a graphical regular representation (GRR), which occupied many authors from 1958 through 1982. In this paper, we review for graph theorists some standard and deep results about finite Frobenius groups, determine classes of finite Frobenius groups and individual groups that do and do not admit GFRs, and classify those Frobenius groups of order at most 300 having a GFR. Because a Frobenius group, as opposed to a regular permutation group, has a highly restricted structure, the GFR problem emerges as algebraically more complex than the GRR problem. This paper concludes with some further questions and a strong conjecture.     

Measurable groups of low dimension

We consider low‐dimensional groups and group‐actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is (finite‐by‐Abelian)‐by‐finite, and that any 2‐dimensional asymptotic group is soluble‐by‐finite. We obtain a field‐interpretation theorem for certain measurable groups, and give an analysis of minimal normal subgroups and socles in groups definable in a supersimple theory of finite rank where infinity is definable. We prove a primitivity theorem for measurable group actions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complex group algebras of finite groups: Brauer's Problem 1

Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m. We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.     

The foundations of enumeration theory for finite nilpotent groups

This paper is the first in a series of papers that lay the foundations of enumeration theory for finite groups including the classical inversion calculus on segments of the natural series and on lattices of subsets of finite sets. Since it became possible to calculate the Möbius function on all subgroups of finite nilpotent groups, the Möbius inversion on these groups began to play a decisive role. The efficiency of the inversion method as a regular technique suitable for solution of enumeration problems of group theory is illustrated with a number of concrete and very important enumerations. Bibliography: 13 titles.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

Pronormal subgroups and homomorphs in finite groups

A local version of the theory of homomorphs and Schunck classes is given. It is shown that for any finite soluble group the pronormal subgroups are precisely the covering subgroups with respect to “Schunck sets” in this group. As an application simple proofs of some results on pronormal subgroups of finite soluble groups are obtained. Finally a question of Doerk is answered in the negative: any finite soluble group is a subgroup of a minimal non-trivial pronormal subgroup of some finite soluble group.     

Independence, measure and pseudofinite fields

We give a measure-theoretic refinement of the Independence Theorem in pseudofinite fields.     

Negative K-Theory of Generalized Quaternion Groups and Binary Polyhedral Groups

Classical group representation theory is used to determine which finite groups have finite negative K-theory. There follows a computation of the K _?1 of integral group rings ZG for all finite non-abelian subgroups of the group SU(2) of unit quaternions. In principle, the method applies to any finite group.     

Brauer–Clifford equivalence of full matrix algebras

The Brauer–Clifford group was introduced to describe the Clifford theory for finite groups. It was proved that it has a natural homomorphism into a Brauer group, and the kernel of this homomorphism is the set of all equivalence classes of G-algebras which are full matrix algebras. In this paper, we prove that this kernel is isomorphic to a second cohomology group. In the Clifford theory for finite groups situation, we characterize families of characters which yield elements in the full matrix subgroup of the Brauer–Clifford group as those where an appropriate character has Schur index one. We also show, in this case, how to compute the element of the second cohomology group associated with this family of characters.     

论有限群元素共轭类的平均长度

本文研究有限群元素共轭类的平均长度问题.利用初等群论方法和有限群特征标理论,在共轭类平均长度为某一定数时,获得了对有限群结构的刻划,且对有限群数量性质的研究是有意义的.     

Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes

This a first step to develop a theory of smooth, étale, and unramified morphisms between Noetherian formal schemes. Our main tool is the complete module of differentials, which is, a coherent sheaf whenever the map of formal schemes is of pseudofinite type. Among our results, we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of Noetherian formal schemes is flat and its module of differentials is locally free. The article closes with a version of Zariski's Jacobian criterion.     

Finite group actions on bordered surfaces of small genus

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2≤p≤6. We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p≥2.     

Connecting monomiality questions with the structure of rational group algebras

The purpose of this paper is to show the vastness of the class of generalized strongly monomial groups and the implication of its study on monomiality questions studied by group theorists. The so called generalized strongly monomial groups, which is a generalization of strongly monomial groups introduced by Olivieri, del Río and Simón, arose in a recent work of authors while understanding the algebraic structure of rational group algebras.     

一类2nm(m为奇数)阶有限群的构造

本文研究了一类2nm(m为奇数)阶有限群的构造,利用解数论同余方程的方法和群的扩张理论等知识,得到了具有奇数m阶循环正规子群、其补子群为循环群的2nm阶有限群的构造及相关的计数定理.     

Orbifold construction for topological field theories

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G?H$ of finite groups assigns in a functorial way to a G-equivariant topological field theory an H-equivariant topological field theory, the pushforward theory. When H is the trivial group, this yields an orbifold construction for G-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

Normal ergodic actions

The von Neumann-Halmos theory of ergodic transformations with discrete spectrum makes use of the duality theory of locally compact abelian groups to characterize those transformations preserving a probability measure, which are defined by a rotation on a compact abelian group. We use the recently developed duality between general locally compact groups and Hopf-von Neumann algebras to characterize those actions of a locally compact group, preserving a σ-finite measure, which are defined by a dense embedding in another group. They are characterized by the property of normality, previously introduced by the author, and motivated by Mackey's theory of virtual groups. The discrete spectrum theory is readily seen to come out as the special case in which the invariant measure is finite.     

Über die Jordansche Verallgemeinerung der Eulerschen Punktion

The Jordan totient revisited: C. Jordan’s generalization of the Euler ø—function is disregarded in most textbooks on number theory and algebra today. We prove some results about this important arithmetical function and demonstrate its usage in finite group theory and complex representation theory. Our interest in this totient was “induced” by studying all complex representations of finite nilpotent Heisenberg groups. The structure of the dual of these groups can be related to a theorem on Jordan totients. Our theorems lead to some peculiar equations involving the greatest common divisor (in German: ggT). We present an interesting example for the interaction of conjugacy classes and equivalence classes of irreducible representations in a special manner.