Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

Abelian subgroup separability of certain generalized free products of free or finitely generated nilpotent groups

In this paper we prove that certain generalized free products of abelian subgroup separable groups, amalgamating an infinite cyclic subgroup, are abelian subgroup separable. Applying this, we derive that tree products of free groups or finitely generated nilpotent groups, amalgamating infinite cyclic subgroups, are abelian subgroup separable.     

Classification of cyclic braces

Etingof, Schedler, and Soloviev have shown [P. Etingof, T. Schedler, A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999) 169-209] that T-structures on cyclic groups come from bijective 1-cocycles and thus give rise to solutions of the quantum Yang-Baxter equation. At the end of their paper, they ask for a classification of T-structures on cyclic groups, especially p-groups. We solve the latter problem by means of generalized radical rings (=braces).     

An analogue of the classical Yang-Baxter equation for general algebraic structures

In this paper, we introduce an analogue of the classical Yang-Baxter equation for general algebraic structures (including nonassociative algebras and vertex operator algebras). Moreover, we give several ways to construct solutions of the equation in case the algebraic structure is graded by an abelian group. In particular, we construct some unitary nondegenerate trignometric solutions of the classical Yang-Baxter equation for affine Lie algebras by means of our equation.This paper was written while the author was a graduate student in the Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, U.S.A.     

On the IYB-property in some solvable groups

A finite group G is called Involutive Yang-Baxter (IYB) if there exists a bijective 1-cocycle ${\chi: G \longrightarrow M}$ for some ${\mathbb{Z}G}$ -module M. It is known that every IYB-group is solvable, but it is still an open question whether the converse holds. A characterization of the IYB property by the existence of an ideal I in the augmentation ideal ${\omega\mathbb{Z}G}$ complementing the set 1?G leads to some speculation that there might be a connection with the isomorphism problem for ${\mathbb{Z}G}$ . In this paper we show that if N is a nilpotent group of class two and H is an IYB-group of order coprime to that of N, then ${N \rtimes H}$ is IYB. The class of groups that can be obtained in that way (and hence are IYB) contains in particular Hertweck’s famous counterexample to the isomorphism conjecture as well as all of its subgroups. We then investigate what an IYB structure on Hertweck’s counterexample looks like concretely.     

Functors between shape categories

Permutation groups of prime power degree are investigated here through the study of the corresponding group algebra of the set of all functions from the underlying set on which the permutation group acts to a finite field of characteristic p. For the case when the permutation group is of degree p² acting on a set consisting of the direct product of two elementary abelian p-groups, the structure of a minimal permutation module is obtained under certain conditions. The proofs do not depend on the recent classification results of finite simple groups.     

Finite quotients of groups of I-type

To every group of I-type, we associate a finite quotient group that plays the role that Coxeter groups play for Artin–Tits groups. Since groups of I-type are examples of Garside groups, this answers a question of D. Bessis in the particular case of groups of I-type. Groups of I-type are related to finite set-theoretical solutions of the Yang–Baxter equation. So, our result provides a new tool to attack the problem of the classification of finite set-theoretical solutions of the Yang–Baxter equation.     

Precosheaves of pro-sets and abelian pro-groups are smooth

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.     

An inverse eigenvalue problem and an associated approximation problem for generalized K-centrohermitian matrices

A partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized K-centrohermitian matrices are considered. It is shown under which conditions the inverse eigenproblem has a solution. An expression of its general solution is given. In case a solution of the inverse eigenproblem exists, the optimal approximation problem can be solved. The formula of its unique solution is given.     

The retraction relation for biracks

In [9] Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution $(X,\sigma ,\tau )$ of the Yang–Baxter equation, the equivalence relation ～ defined on the set X and they considered a new non-degenerate involutive induced retraction solution defined on the quotient set $X^{～}$. It is well known that translating set-theoretical non-degenerate solutions of the Yang–Baxter equation into the universal algebra language we obtain an algebra called a birack. In the paper we introduce the generalized retraction relation ≈ on a birack, which is equal to ～ in an involutive case. We present a complete algebraic proof that the relation ≈ is a congruence of the birack. Thus we show that the retraction of a set-theoretical non-degenerate solution is well defined not only in the involutive case but also in the case of all non-involutive solutions.     

Block-iterative methods for consistent and inconsistent linear equations

Summary We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.     

Abstract simplicity of complete Kac-Moody groups over finite fields

Let G be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix A, as constructed by Tits. It is known that G admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group which is defined to be the closure of G in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices A of rank at least four. Our proof uses Tits’ simplicity theorem for groups with a BN-pair and methods from the theory of pro-p groups.     

A structure theorem for abelian quasi-ordered groups

We introduce a notion of compatible quasi-ordered groups which unifies valued and ordered abelian groups. It was proved by S.M. Fakhruddin that a compatible quasi-order on a field is always either an order or a valuation. We show here that the group case is more complicated than the field case and describe the general structure of a compatible quasi-ordered abelian group. We then define a notion of Hahn product of compatible quasi-ordered groups and generalize Hahn's embedding theorem to quasi-ordered groups. We also develop a notion of quasi-order-minimality and establish a connection with C-minimality, thus answering a question of F. Delon. Finally, we use compatible quasi-ordered groups to give an example of a C-minimal group which is neither an ordered nor a valued group.     

Finite presentation of fibre products of metabelian groups

We show that if Γ is a finitely presented metabelian group, then the “untwisted” fibre product or pull-back P associated to any short exact sequence 1→N→Γ→Q→1 is again finitely presented. In contrast, if N and Q are abelian, then the analogous “twisted” fibre-product is not finitely presented unless Γ is polycyclic. Also a number of examples are constructed, including a non-finitely presented metabelian group P with finitely generated.     

Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.     

Explicit determination of certain minimal constabelian codes

Constabelian codes can be viewed as ideals in twisted group algebras over finite fields. In this paper we study decomposition of semisimple twisted group algebras of finite abelian groups and prove results regarding complete determination of a full set of primitive orthogonal idempotents in such algebras. We also explicitly determine complete sets of primitive orthogonal idempotents of twisted group algebras of finite cyclic and abelian p-groups. We also describe methods of determining complete set of primitive idempotents of abelian groups whose orders are divisible by more than one prime and give concrete (numerical) examples of minimal constabelian codes, illustrating the above mentioned results.     

Strong Morita equivalence of higher-dimensional noncommutative tori. II

We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K ₀-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, held by George A. Elliott.