Finite General Groups Research Articles

Non-cancellation Phenomena in the Class Of Finitely Generated Groups with Finite Commutator Subgroups

We observe some non-cancellation phenomena arising from recent studies of the non-cancellation set χ (G) of a certain group G. Here χ (G) denotes the set of all isomorphism classes of groups H such that H × Z G × Z. In particular, there is a group structure on the set χ (G), and we compare this group operation with direct product formation. We also study a particular class of groups for which the non-cancellation set is trivial.

Neighbourhood Graphs of Cayley Graphs for Finitely-generated Groups

In this short note the neighbourhood graph of a Cayley graph is considered. It has, as nodes, a symmetric generating set of a finitely-generated group Γ. Two nodes are connected by an edge if one is obtained from the other by multiplication on the right by one of the generators. Two necessary conditions on the graphs are shown. One is a condition on the degrees of the graph, the other concerns complete subgraphs.

Automorphisms of Free Groups and Outer Space

This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations.

Uniform Growth in Groups of Exponential Growth

This is an exposition of examples and classes of finitely-generated groups which have uniform exponential growth. The main examples are non-Abelian free groups, semi-direct products of free Abelian groups with automorphisms having an eigenvalue of modulus distinct from 1, and Golod–Shafarevich infinite finitely-generated p-groups. The classes include groups which virtually have non-Abelian free quotients, nonelementary hyperbolic groups, appropriate free products with amalgamation, HNN-extensions and one-relator groups, as well as soluble groups of exponential growth. Several open problems are formulated.

A Krengel-type theorem for finitely-generated nilpotent groups

A Krengel-type theorem for finitely-generated nilpotent groups

Source algebra equivalences for blocks¶of finite general linear groups over a fixed field

T. Jost proved that Donovan's conjecture holds for the unipotent blocks of the finite general linear groups over a fixed field. In this paper, we show that the Morita equivalences exhibited by Jost are in fact equivalences between the source algebras of the corresponding blocks, and thus that Puig's conjecture holds for the unipotent blocks of finite general groups over a fixed field.

Asymptotic Cones of Finitely Generated Groups

Asymptotic Cones of Finitely Generated Groups

On the asymptotic geometry of abelian-by-cyclic groups

Gromov’s Polynomial Growth Theorem [Gro81] states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups. Gromov’s theorem inspired the more general problem (see, e.g. [GdlH91]) of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure—in short, are solvable groups quasi-isometrically rigid? In general they aren’t: very recently A. Dioubina [Dio99] has found a solvable group which is quasi-isometric to a group which is not virtually solvable; these groups are finitely generated but not finitely presentable. In the opposite direction, first steps in identifying quasi-isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples, the solvable BaumslagSolitar groups, in [FM98] and [FM99b]. The goal of the present paper is to show that a much broader class of solvable groups, the class of finitely-presented, nonpolycyclic, abelian-bycyclic groups, is characterized among all finitely-generated groups by its quasi-isometry type. We also give a complete quasi-isometry classification of the groups in this class; such a classification for nilpotent groups remains a major open question. Motivated by these results, we offer a conjectural picture of quasi-isometric classification and rigidity for polycyclic abelianby-cyclic groups in §10.1.

Determination of Endomorphisms of Free Metabelian Lie Algebras

Answering V. E. Shpil′răin’s question [1, Problem 13.66], E. I. Timoshenko [2] found necessary and sufficient conditions on elements of a free finitely-generated metabelian group G of rank n for every endomorphism of G to be uniquely determined from its action on these elements. Let Mn be a free metabelian Lie algebra with n free generators. The purpose of this article consists in clarifying necessary and sufficient conditions on elements b1, b2, . . . , bm of the Lie algebra Mn for every endomorphism φ of this algebra to be uniquely determined from its actions on these elements, or shortly the conditions for b1, b2, . . . , bm to determine each endomorphism of Mn uniquely. Let K be an arbitrary field and let L be a Lie algebra over K. Denote the universal enveloping algebra for L by UL. It is well known that UL is an integral domain. Denote by F the free Lie algebra over K with the set of free generators {y1, . . . , yn}. The universal enveloping algebra of F is a free associative algebra with the same set of free generators. The ideal generated by F in UF is a free right UF -module with basis y1, . . . , yn. Therefore, each element f ∈ FUF has the unique expansion

Designing local orthogonal bases on finite groups I: Abelian case

We extend to general finite groups a well-known relation used for checking the orthogonality of a system of vectors as well as for orthogonalizing a nonorthogonal one. This, in turn, is used for designing local orthogonal bases obtained by unitary transformations of a single prototype filter. The first part of this work considers abelian groups. The second part considers nonabelian groups where, as an example, we show how to build such bases where the group of unitary transformations consists of modulations and rotations. These bases are useful for building systems for evaluating image quality.

Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II

Let BS(1,n)= < a,b: aba^{-1}=b^n >. We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.

On the Complexity of the Isomorphism Relation for Finitely Generated Groups

Confirming a conjecture of G. Hjorth and A. Kechris (1996, Ann. Pure Appl. Logic82, 221–272) we prove that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group F2 on two generators.

Generic and $q$-Rational Representation Theory

Part I of this paper develops various general concepts in generic representation and cohomology theories. Roughly speaking, we provide a general theory of orders in non-semisimple algebras applicable to problems in the representation theory of finite and algebraic groups, and we formalize the notion of a “generic” property in representation theory. Part II makes new contributions to the non-describing representation theory of finite general linear groups. First, we present an explicipt Morita equivalence connecting GL_n(q) with the theory of q -Schur algebras, extending a unipotent block equivalence of Takeuchi [T]. Second, we apply this Morita equivalence to study the cohomology groups H^*(GL_n(q),L) , when L is an irreducible module in non-describing characteristic. The generic theory of Part I then yields stability results for various groups H^t(GL_n(q),L) , reminscent of our general theory [CPSK] with van der Kallen of generic cohomology in the describing characteristic case, (in turn, the stable value of such a cohomology group can be expressed in terms of the cohomology of the affine Lie algebra \widehat{\mathfrak{gl}}_n(ℂ) .) The arguments entail new applications of the theory of tilting modules for q -Schur algebras. In particular, we obtain new complexes involving tilting modules associated to endomorphism algebras obtained from general finite Coxeter groups.

Digraphical Regular Representations of Infinite Finitely Generated Groups

A directed Cayley graphXis called a digraphical regular representation (DRR) of a groupGif the automorphism group ofXacts regularly onX. LetSbe a finite generating set of the infinite cyclic groupZ. We show that a directed Cayley graphX(Z,S) is aDRRofZif and only ifS≠S−1. IfX(Z,S) is not aDRRwe show thatAut(X(Z,S)) =D∞. As a general result we prove that a Cayley graphXof a finitely generated torsion-free nilpotent groupNis aDRRif and only if no non-trivial automorphism ofNof finite order leaves the generating set invariant.

Escaping orbits in trace maps

We study the finitely-generated group A of invertible polynomial mappings from C 3 to itself (or R 3 to itself) which preserve the Fricke-Vogt invariant I( x, y, z) = x 2 + y 2 + z 2 − 2 xyz − 1. Using properties of suitably-chosen generators, we give a necessary condition and sufficient conditions for infinite order elements of A to have an unbounded orbit escaping to infinity in forward or backward time. Our main motivation for this study is that A includes the so-called trace maps derived from transfer matrix approaches to various physical processes displaying non-periodicity in space or time. As shown previously, characterising escaping orbits leads to various conclusions for the physical model and vice versa (e.g. electronic properties of ID quasicrystals). Our results generalize in a simple constructive way those previously proved for Fibonacci-type trace maps.

Powers in Finitely Generated Groups

In this paper we study the set Γ n \Gamma ^n of n t h n^{th} -powers in certain finitely generated groups Γ \Gamma . We show that, if Γ \Gamma is soluble or linear, and Γ n \Gamma ^n contains a finite index subgroup, then Γ \Gamma is nilpotent-by-finite. We also show that, if Γ \Gamma is linear and Γ n \Gamma ^n has finite index (i.e. Γ \Gamma may be covered by finitely many translations of Γ n \Gamma ^n ), then Γ \Gamma is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the S S -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.

Character Variety of Representations of a Finitely Generated Group in SL2

Character Variety of Representations of a Finitely Generated Group in SL2

Factoring by subsets of cardinality of prime power

Redei's theorem asserts that if a finite abelian group is expressed as a direct product of subsets of prime cardinality, then at least one of the factors must be periodic. (A periodic subset is a direct product of some subset and a nontrivial subgroup.) A. D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality that is a power of a prime, then at least one of the factors is periodic. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subsets of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.

A novel approach for constructing symmetry‐adapted basis sets for quantum‐chemical calculations. I. Real symmetry‐adapted orbitals

AbstractA novel method for generating symmetry‐adapted basis orbitals (widely applied in quantum chemistry) is proposed. It extensively combines the ideas of both the local‐group and projection‐operator methods. Some important improvements of the previous formalisms are made, namely: (i) the simple procedure for the construction of the transformation matrix (TM) while the matrix making this TM unitary is extracted individually for further usage, (ii) the partial reduction of the initial reducible representation results in the possibility of conserving all matrices real even in the cases when the group in question has complex irreducible representations. The procedure proposed in this paper is applied to the general finite group which could be both the usual point or large unit cell group. Our method is well suited for computer utilization.

An extension for Hajós' theorem

Hajós theorem asserts that if a finite abelian group is expressed as a direct product of cyclic subsets of prime cardinality, then at least one of the factors must be a subgroup. (A cyclic subset is a ‘front end’ of a cyclic subgroup.) A.D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality a power of a prime, then at least one of the factors is a direct product of some subset and a nontrivial subgroup. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subset of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.