On automorphisms of finite simple Chevalley groups

The paper deals with the splitting properties of the automorphism groups of finite Chevalley groups. Using the action of symmetries of a Dynkin diagram on the corresponding Weyl group, a sufficient condition is developed for the existence of a complement for the inner automorphism group in the automor ph-ism group of a finite Chevalley group. The condition is verified for Chevalley groups of the classical types (viz., the group of type A_l, B_l, C_l,and D_l) as well as the exceptional groups of type E₆ and E₇, under suitable restrictions on the base fields.     

Chapter 1. Extension of the fundamental theorem of finite semigroups

This paper proves that some useful commutivity relations exist among semigroup wreath product factors that are either groups or combinatorial “units” U₁, U₂, or U₃. Using these results it then obtains some characterizations of each of the classes of semigroups buildable from U₁'s, U₂'s, and groups (“buildable” meaning “dividing a wreath product of”).We show that up to division U₁'s can be moved to the right and U₂'s, and groups to the left over other units and groups, if it is allowed that the factors involved be replaced by their direct products, or in the case of U₂, even by a wreath product. From this it is deduced that U₁'s and U₂'s do not affect group complexity, that any semigroup buildable from U₁'s, U₂'s, and groups has group complexity 0 or 1, and that all such semigroups can be represented, up to division, in a canonical form—namely, as a wreath product with all U₁'s on the right, all U₂'s on the left, and a group in the middle. This last fact is handy for developing charactérizations.An embedding theorem for semigroups with a unique 0-minimal ideal is introduced, and from this and the commutivity results and some constructions proved for RLM semigroups, there is obtained an algebraic characterization for each class of semigroups that is a wreath product-division closure of some combination of U₁'s, U₂'s, and the groups. In addition it is shown, for i = 1,2,3, that if the unit U_i does not divide a semigroup S, then S can be built using only groups and units not containing U_i. Thus, it can be deduced that any semigroup which does not contain U₃ must have group complexity either 0 or 1. This then establishes that indeed U₃ is the determinant of group complexity, since it is already proved that both U₁ and U₂ are transparent with regard to the group complexity function, and it is known that with U₃ (and groups) one can build semigroups with complexities arbitrarily large. Another conclusion is a combinatorial counterpart for the Krohn-Rhodes prime decomposition theorem, saying that any semigroups can be built from the set of units which divide it together with the set of those semigroups not having unit divisors. Further, one can now characterize those semigroups which commute over groups, showing a semigroup commutes to the left over groups iff it is “R₁” (i.e., does not contain U₁, i.e., is buildable form U₂'s and groups), and commutes to the right over groups iff it does not contain U₂ (i.e., is buildable from groups and U₁'s). Finally, from the characterizations and their proofs one sees some ways in which groups can do the work of combinatorials in building combinatorial semigroups.     

Surface framed braids

In this paper we introduce the framed pure braid group on n strands of an oriented surface, a topological generalisation of the pure braid group P _n . We give different equivalent definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of P _n , usually called surface pure braid groups. The notion of surface framed braid groups is also introduced.     

On the groups of unitriangular automorphisms of relatively free groups

We describe the structure of the group U _n of unitriangular automorphisms of the relatively free group G _n of finite rank n in an arbitrary variety C of groups. This enables us to introduce an effective concept of normal form for the elements and present U _n by using generators and defining relations. The cases n = 1, 2 are obvious: U ₁ is trivial, and U ₂ is cyclic. For n ?? 3 we prove the following: If G _n?1 is a nilpotent group then so is U _n . If G _n?1 is a nilpotent-by-finite group then U _n admits a faithful matrix representation. But if the variety C is different from the variety of all groups and G _n?1 is not nilpotent-by-finite then U _n admits no faithful matrix representation over any field. Thus, we exhaustively classify linearity for the groups of unitriangular automorphisms of finite rank relatively free groups in proper varieties of groups, which complements the results of Olshanskii on the linearity of the full automorphism groups AutG _n . Moreover, we introduce the concept of length of an automorphism of an arbitrary relatively free group G _n and estimate the length of the inverse automorphism in the case that it is unitriangular.     

A discrete homotopy theory for binary reflexive structures

We present a simple combinatorial construction of a sequence of functors σ_k from the category of pointed binary reflexive structures to the category of groups. We prove that if the relational structure is a poset P then the groups are (naturally) isomorphic to the homotopy groups of P when viewed as a topological space with the topology of ideals, or equivalently, to the homotopy groups of the simplicial complex associated to P. We deduce that the group σ_k(X,x₀) of the pointed structure (X,x₀) is (naturally) isomorphic to the kth homotopy group of the simplicial complex of simplices of X, i.e. those subsets of X which are the homomorphic image of a finite totally ordered set.     

A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D ₄, F ₄ and H ₄. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the twodimensional non-crystallographic Coxeter groups I ₂(n) and the threedimensional groups A ₃, B ₃, as well as the icosahedral group H ₃. IPPP/12/49, DCPT/12/98     

Abelian groups with a vanishing homology group

An abelian group A is called absolutely abelian, if in every central extension N ? G ? A the group G is also abelian. The abelian group A is absolutely abelian precisely when the Schur multiplicator H₂A vanished. These groups, and more generally groups with H_nA = 0 for some n, are characterized by elementary internal properties. (Here H₁A denotes the integral homology of A.) The cases of even n and odd n behave strikingly different. There are 2^?ο different isomorphism types of abelian groups A with reduced torsion subgroup satisfying H_2nA = 0. The major tools are direct limit arguments and the Lyndon-Hochschild-Serre (L-H-S) spectral sequence, but the treatment of absolutely abelian groups does not use spectral sequences. All differentials d^r for r ≥ 2 in the L-H-S spectral sequence of a pure abelian extension vanish. Included is a proof of the folklore theorem, that homology of groups commutes with direct limits also in the group variable, and a discussion of the L-H-S spectral sequence for direct limits.     

Recognizing Finite Groups Through Order and Degree Patterns

The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns:All sporadic simple groups,the alternating groups Ap(p≥5 is a twin prime)and some simple groups of the Lie type.In this paper,the authors continue this investigation.In particular,the authors show that the symmetric groups Sp+3,where p+2 is a composite number and p+4 is a prime and 97相似文献   

Conservative groups,indicability, and a conjecture of Howie

The class of locally p-indicable groups coincides with the class of $\text{Z}$_p conservative groups. Howie's conjecture is valid for locally p-indicable groups.     

A product of varieties in relation to Baer invariants

The concept of the Baer invariant is useful in classifying groups into isologism classes. In this paper two sequences of varieties {^rV_n}n∈N₀ and {^lV_n}n∈N₀, are considered from a given variety V. The structure of Baer invariants of some groups with respect to these varieties, are determined for some specific V.     

On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

The spectrum of a finite group is the set of its element orders. Two groups are isospectral whenever they have the same spectra. We consider the classes of finite groups isospectral to the simple symplectic and orthogonal groups B ₃(q), C ₃(q), and D ₄(q). We prove that in the case of even characteristic and q > 2 these groups can be reconstructed from their spectra up to isomorphisms. In the case of odd characteristic we obtain a restriction on the composition structure of groups of this class.     

The π-π-theorem for manifold pairs with boundaries

The surgery obstruction of a normal map to a simple Poincaré pair (X, Y) lies in the relative surgery obstruction group L _*(π ₁(Y) → π ₁(X)). A well-known result of Wall, the so-called π-π-theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincaré pair with π ₁(X) ? π ₁(Y) is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced the surgery obstruction groups LP _* for manifold pairs and splitting obstruction groups LS _*. In the present paper, we formulate and prove for manifold pairs with boundary results similar to the π-π-theorem. We give direct geometric proofs, which are based on the original statements of Wall’s results and apply obtained results to investigate surgery on filtered manifolds.     

Problems in algebra inspired by universal algebraic geometry

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant K _Θ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H ₁ and H ₂ have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) K _Θ(H ₁) and K _Θ(H ₂) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ⁰ be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ⁰) for different varieties Θ and also the groups of autoequivalences of Θ⁰. The problem is to describe these groups for different Θ.     

Bianchi groups are conjugacy separable

We prove that non-uniform arithmetic lattices of SL₂(C) and consequently the Bianchi groups are conjugacy separable. The proof is based on recent deep results of Agol, Long, Reid and Minasyan. The conjugacy separability of groups commensurable with limit groups is also established.     

OD-characterization of all finite nonabelian simple groups with orders having prime divisors at most 13

The main purpose of this article is to determine h _OD (M) for every finite nonabelian simple group M with order having prime divisors at most 13. This result is an analog of the result by A. V. Vasil’ev [1] about the recognizability of these simple groups by spectrum (the set of element orders). By the available results, we need only consider the groups L ₆(3), U ₄(5), G ₂(4), L ₅(3), S ₄(8), U ₆(2), and O ₈ ⁺ (3).     

Approximation algorithms for group prize-collecting and location-routing problems

In this paper we develop approximation algorithms for generalizations of the following three known combinatorial optimization problems, the Prize-Collecting Steiner Tree problem, the Prize-Collecting Travelling Salesman Problem and a Location-Routing problem.Given a graph G=(V,E) on n vertices and a length function on its edges, in the grouped versions of the above mentioned problems we assume that V is partitioned into k+1 groups, {V₀,V₁,…,V_k}, with a penalty function on the groups. In the Group Prize-Collecting Steiner Tree problem the aim is to find S, a collection of groups of V and a tree spanning the rest of the groups not in S, so as to minimize the sum of the costs of the edges in the tree and the costs of the groups in S. The Group Prize-Collecting Travelling Salesman Problem, is defined analogously. In the Group Location-Routing problem the customer vertices are partitioned into groups and one has to select simultaneously a subset of depots to be opened and a collection of tours that covers the customer groups. The goal is to minimize the costs of the tours plus the fixed costs of the opened depots. We give a -approximation algorithm for each of the three problems, where I is the cardinality of the largest group.     

Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E ₈, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A ₁ × A ₁ × A ₁, A ₃, B ₃ and H ₃ of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A ₁ × A ₁ × A ₁, A ₃, B ₃ and H ₃ yield a new simple construction of binary polyhedral groups. These in turn generate the groups A ₁ × A ₁ × A ₁ × A ₁, D ₄, F ₄ and H ₄ of rank 4 and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D ₄, F ₄ and H ₄ are exceptional structures, which our study sheds new light on. IPPP/12/26, DCPT/12/52