A little more group theory

Introduction I. A Theory of Groups and Organizations A. The purpose of organization B. Public goods and large groups C. The traditional theory of groups D. Small groups E. Exclusive and inclusive groups F. A taxonomy of groups II. Group Size and Group Behavior A. The coherence and effectiveness of small groups B. Problems of the traditional theories C. Social incentives and rational behavior III. The Labor Union and Economic Freedom A. Coercion in labor unions B. Labor-union growth in theory and practice C. The closed shop and economic freedom in the latent group D. Government intervention and economic freedom in the latent group IV. Orthodox Theories of State and Class A. The economists' theory of the state B. The Marxian theory of state and class C. The logic of the Marxian theory V. Orthodox Theories of Pressure Groups A. The philosophical view of pressure groups B. Institutional economics and the pressure group--John R. Commons C. Modern theories of pressure groups--Bentley, Truman, Latham D. The logic of group theory VI. The By-Product and Special Interest Theories A. The by-product theory of large pressure groups B. Labor lobbies C. Professional lobbies D. The special interest theory and business lobbies E. Government promotion of political pressure F. Farm cooperatives and farm lobbies G. Noneconomic lobbies H. The forgotten groups--those who suffer in silence Index

The object of this paper is to bring together several points connected with the general theory of correspondences and continuous groups, and to apply them to the theory of screws. Although the several results are in general not new, it seems of interest to give the accompanying presentation of the subject, as it furnishes an excellent examiple of the way in which the theory of continuous groups underlies the whole theory of correspondences. t The first section is devoted to general theory. Use is made of the theorem of LIE j that if we have a continuous group in n variables together with an invariant equation system inlvolving m paramneters, then a, group of the parameters' exists which is isomorphic with the given group, and it is pointed out that this theorem is fundamental iii all correspondences. ? The correspondence established is that between a P,M and a P n Contact transformationl is the particular case when mn n. The screw geometry is developed from the projective group in three dimensions together with the system of equations which define a general straight line. The general theory leads at once to two important results in conniection with the theory of groups: 1) The genercal continuous confornmal group in fourc dimensions is simply isomorphic with the generalprojective group in three. 2) Both these groups are simply isomorphic with the continuous projective group infive dinmensions which leaves a giv'en quadric invariant. There follows an imnmediate generalization of part of the second theorem. We have in fact the following: 3) The general conformal group in space of n dimensions is simply isomorphic with the projective group in space of n + 1 dimensions which preserves a given quadric. These three results are due to KLEIN. Some slight differences appear

It is generally believed (and for the most part it is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this article we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups Un), and under a certain hypothesis relating the characteristic p to both n and the dimension d of a representation (specifically, p ≥ max(n, 2d)), Lie theory is completely sufficient to determine the representation theories of these groups. To finish, we mention some important analogies (both functorial and cohomological) between the characteristic zero theories of these groups and their “generic” representation theory in characteristic p.

The notion of n-ary group is a generalization of the binary group so many of the results from the theory of groups have n-ary analogue in theory of n-ary groups. But there are significant differences in these theories. For example, multiplier of the direct product of n-ary groups does not always have isomorphic copy in this product (in paper there is an example). It is proved that the direct product ∏ i∈I ⟨Ai , fi⟩ n-ary groups has n-ary subgroup isomorphic to ⟨Aj , fj ⟩ (j ∈ I), then and only when there is a homomorphism of ⟨Aj , fj ⟩ in ∏ i∈I,i=j ⟨Ai , fi⟩. Were found necessary and sufficient conditions for in direct product of n-ary groups, each of the direct factors had isomorphic copy in this product and the intersection of these copies singleton (as well as in groups) – each direct factor has a idempotent. For every n-ary group, can define a binary group which helps to study the n-ary group, that is true Gluskin-Hossu theorem: for every n-ary group of ⟨G, f⟩ for an element e ∈ G can define a binary group ⟨G, ·⟩, in which there will be an automorphism φ(x) = f(e, x, cn−2 1 ) and an element d = f( (n) e ) such that the following conditions are satisfied: f(x n 1 ) = x1 · φ(x2) · . . . · φ n−1 (xn) · d, x1, x2, . . . , xn ∈ G; (4) φ(d) = d; (5) φ n−1 (x) = d · x · d −1 , x ∈ G. (6) Group ⟨G, ·⟩, which occurs in Gluskin-Hossu theorem called retract n-ary groups ⟨G, f⟩. Converse Gluskin-Hossu theorem is also true: in any group ⟨G, ·⟩ for selected automorphism φ and element d with the terms (5) and (6), given n-ary group ⟨G, f⟩, where f defined by the rule (4). A n-ary group called (φ, d)- defined on group ⟨G, ·⟩ and denote derφ,d⟨G, ·⟩. Was found connections between n-ary group, (φ, d)-derived from the direct product of groups and n-ary groups that (φi , di)-derived on multipliers of this product: let ∏ i∈I ⟨Ai , ·i⟩ – direct product groups and φi , di – automorphism and an element in group ⟨Ai , ·i⟩ with the terms of (5) and (6) for any i ∈ I. Then derφ,d ∏ i∈I ⟨Ai , ·i⟩ = ∏ i∈I derφi,di ⟨Ai , ·i⟩, where φ – automorphism of direct product of groups ∏ i inI ⟨Ai , ·i⟩, componentwise given by the rule: for every a ∈ ∏ i∈I Ai , φ(a)(i) = φi(a(i)) (called diagonal automorphism), and d(i) = di for any i ∈ I. In the theory of n-ary groups indecomposable n-ary groups are finite primary and infinite semicyclic n-ary groups (built by Gluskin-Hossu theorem on cyclic groups). We observe n-ary analogue indecomposability cyclic groups. However, unlike groups, finitely generated semi-abelian n-ary group is not always decomposable into a direct product of a finite number of indecomposable semicyclic n-ary groups. It is proved that any finitely generated semiabelian n-ary group is isomorphic to the direct product finite number of indecomposable semicyclic n-ary groups (infinite or finite primary) if and only if in retract this n-ary group automorphism φ from Gluskin-Hossu theorem conjugate to some diagonal automorphism.

The theory of Leibniz algebras has been developing quite intensively. Most of the results on the structural features of Leibniz algebras were obtained for finite-dimensional algebras and many of them over fields of characteristic zero. A number of these results are analogues of the corresponding theorems from the theory of Lie algebras. The specifics of Leibniz algebras, the features that distinguish them from Lie algebras, can be seen from the description of Leibniz algebras of small dimensions. However, this description concerns algebras over fields of characteristic zero. Some reminiscences of the theory of groups are immediately striking, precisely with its period when the theory of finite groups was already quite developed, and the theory of infinite groups only arose, i.e., with the time when the formation of the general theory of groups took place. Therefore, the idea of using this experience naturally arises. It is clear that we cannot talk about some kind of similarity of results; we can talk about approaches and problems, about application of group theory philosophy. Moreover, every theory has several natural problems that arise in the process of its development, and these problems quite often have analogues in other disciplines. In the current survey, we want to focus on such issues: our goal is to observe which parts of the picture involving a general structure of Leibniz algebras have already been drawn, and which parts of this picture should be developed further.

A good deal of research in C∗-algebra K -theory in recent years has been devoted to the Baum-Connes conjecture [3], which proposes a formula for the K -theory of group C∗-algebras that blends group homology with the representation theory of compact subgroups. The conjecture has brought C∗algebra theory into close contact with manifold theory through its obvious similarity to the Borel conjecture of surgery theory [9,31] and its links with the theory of positive scalar curvature [27]. In addition there are points of contact with harmonic analysis, particularly the tempered representation theory of semisimple groups, although the proper relation between the Baum-Connes conjecture and representation theory is not well understood. The conjecture is most easily formulated for groups which are discrete and torsion-free. For such a group G there is a natural homomorphism

Synthesis of Assur groups via group and matroid theory

Linear Groups: The Accent on Infinite Dimensionality explores some of the main results and ideas in the study of infinite-dimensional linear groups. The theory of finite dimensional linear groups is one of the best developed algebraic theories. The array of articles devoted to this topic is enormous, and there are many monographs concerned with matrix groups, ranging from old, classical texts to ones published more recently. However, in the case when the dimension is infinite (and such cases arise quite often), the reality is quite different. The situation with the study of infinite dimensional linear groups is like the situation that has developed in the theory of groups, in the transition from the study of finite groups to the study of infinite groups which appeared about one hundred years ago. It is well known that this transition was extremely efficient and led to the development of a rich and central branch of algebra: Infinite group theory. The hope is that this book can be part of a similar transition in the field of linear groups. Features This is the first book dedicated to infinite-dimensional linear groups This is written for experts and graduate students specializing in algebra and parallel disciplines This book discusses a very new theory and accumulates many important and useful results

Sylow's theorem which plays a most important part in the theory of groups of a finite order has been recently extended to abstract infinite' and topological groups.2 The definition of Sylow's subgroups is essentially connected with the notion of the order of element, which is fundamental in the whole theory of periodical groups. The generalisation of this notion of the order of element naturally gives rise to new problems in the theory of groups, and in particular to theorems similar to Sylow's theorem which for finite groups bear new results. The results mentioned above concerning abstract infinite groups present one of the consequences of this generalisation of Sylow's theorem. It evengives the possibility of stating a sufficient condition of conjugation of an infinite set of all Sylow's subgroups of an infinite group. In ??1-2 we give Definitions 1-6 and Lemmas 1-10 for any set of prime numbers 7r, whereas for the main results of this paper (Theorems 3-5) it is sufficient to consider a 7r, consisting of a prime number p, but this would not simplify the argument essentially. ?1. Let 7r be a finite or an infinite set of prime numbers, and let 'r denote the set of positive integers each of which, except 1, is the product of a finite number of (positive) powers of distinct prime numbers belonging to 7r. Let 5 be a group with whose elements and subgroups we shall be concerned. Let A = (a), B=(f), C = (y) be sets of indices. Let DA = (D.), X DB = (b#), C= (y), 3A = (ta) denote sets of subgroups of (M. The usual notations of E, c, U, n of set theory will be used. DEFINITION 1. Let an element a of 65 be called a 17r; 1SA I -element if for every a e A, there exists an integer 6a e 7f such that aa e . Let a subgroup 1 be called a 1 ir; !A I -subgroup if all its elements are 1r; 1A l-elements. Clearly if a is a 17r; 1SA I -element, then the cyclic group $ = (a) is a I|; -)A |subgroup. DEFINITION 2. Let us call the set of all 17r; 1'TA I -elements of 6 the 1 r; 1'TA I -set of 65 and denote it by V(ir; 1STA; 65). Thus a I Tr; 'A I-subgroup $ of 65 is one contained in V(ir; DA; 65), and hence LEMMA 1. Any subgroup T of a | r; ?'A I -subgroup $3 is a (7r; ?'A) -subgroup. As 7rf is a multiplicative set, i.e. if 3, 6' E 7rf, then b6' E 7rf, we have

Piaget's demonstration interview technique was used to test the level of development of the thinking capacities of socially dissonant children in a disadvantaged area. His theory of groups and groupings were taken as a frame of reference in interpreting the results, which showed a severe developmental lag in these children. The findings are discussed in terms of effects of cognitive patterns on social adaptation in urban areas.

Infra-red radiation from an engine cylinder

It is well known that in the theory of abstract groups an important role is played by the free groups of which the quotient groups make up an inexhaustible resource for the abstract groups. There exist, however, some more vast classes of abstract groups which contain the free groups as very particular case; they are the classes of P-groups. Any abstract multiplicative group can, as we know, be given by a set A = {a i }, i ∈ I, of generators and an exhaustive set F of defining relations which connect these generators. Any relation between the elements of A results from the relations F and the trivial relations between the elements of A. There exist some properties named P-properties which can be common to all relations, including the trivial relations connecting the elements of certain sets of generators of multiplicative groups. At present, we know about thirty such properties, to each of them corresponds a vast class of P-groups which yields to an elegant general theory. Here, we present some important classes of P-groups, especially the quasi free groups, the quasi free groups modulo n, the quasi free groups moduli N, the free groups modulo n, the free groups moduli N and the P-symmetric groups. Next, we define the P-products of groups, products presenting some analogies with the free product, and we indicate some properties of these products. Then, we introduce the fundamental and the quasi fundamental groups as well as their bases and we define the rank, the essential invariant, of a fundamental group. To conclude, we review some curious properties of two fundamental groups of rank 2, the first is given by a couple of generators bound by an exhaustive set of two defining relations and the second is given by a couple of generators bound by a single defining relation.

It is known that the theory of abelian groups has a model companion but that the theory of groups does not. We show that for any fixed n ≥ 2 n \geq 2 the theory of groups solvable of length ≤ n \leq n has no model companion. For the metabelian case ( n = 2 ) (n = 2) we prove the stronger result that the classes of finitely generic, infinitely generic, and existentially complete metabelian groups are all distinct. We also give some algebraic results on existentially complete metabelian groups.

This paper provides an account of results and methods from the theory of infinite groups admitting only finitely many normalizers of subgroups with a given property. Some new statements on this subject are also proved.

The topics of this conference all in some way evolved from the classical theory of real and complex Lie groups. Indeed, one of the important mathematical goals during the 1950's was to find analogs of the semisimple Lie groups of exceptional type over arbitrary fields. Chevalley completed the first crucial step by producing his famous basis theorem for simple complex Lie algebras, and later Steinberg succeeded in describing these analogs group-theoretically. An important development due to Tits was the theory of groups with a BN -pair and invented buildings; these buildings belong to arbitrary Chevalley groups as naturally as the projective spaces belong to the special linear groups. Since then the theories of algebraic groups and of buildings developed into various directions. However, due to their common origin both theories often lead naturally to similar questions which were attacked by completely different means. In the context of this conference, the PhD thesis by Bernhard Mühlherr and the work by Aloysius Helminck and coauthors on involutions of algebraic groups illustrate this in a quite remarkable way. Both projects contributed strongly to the understanding of the geometry of involutions of algebraic groups, but surprisingly each one has gone unnoticed by the researchers of the other until recently. One of the main objectives of this conference was to bring these two theories closer to each other. The first two lectures on Monday morning familiarised all participants with the concept of buildings; Pierre-Emmanuel Caprace explained the foundations of buildings from the simplicial point of view, while Bernhard Mühlherr introduced the chamber system approach to buildings and explained the power of filtrations when studying sub-geometries of (twin) buildings that arise from the action of certain subgroups of the isometry group of the (twin) building. As these two lectures were of an introductory nature and since their content is already well documented (we refer to the recently published book by Abramenko and Brown for the theory of buildings and to the contributions of Alice Devillers and of Hendrik Van Maldeghem to the Oberwolfach report 20/2008 for an account on filtrations and their powerful applications), we do not include acstracts of these lectures. On Monday afternoon and on Friday morning Aloysius Helminck presented the theory of involutions of algebraic groups, while in Monday's final lecture Max Horn showed how to combine Helminck's theory with M"uhlherr's PhD thesis in order to obtain general and powerful results on the geometry of involutions of groups with a root group datum, a class of groups that contains the semisimple algebraic groups, the split Kac–Moody groups, and the split finite groups of Lie type. Most of Tuesday and part of Wednesday were focussed on the Tits centre conjecture. In a series of two lectures Gerhard Röhrle and Michael Bate presented an algebraic-group approach towards proving the conjecture, while on Tuesday afternoon Katrin Tent presented a combinatorial approach and on Wednesday morning Linus Kramer reported on metric considerations in the context of the Tits centre conjecture. It is our impression that these four lectures have triggered additional activity towards proving the centre conjecture, and that one or more of these approaches will be successful in the near future. The fourth talk on Tuesday afternoon was given by Yiannis Sakellaridis on spherical varieties and automorphic forms, while the second talk on Wednesday by Lizhen Ji presented compactifications of locally symmetric spaces. Thursday's talks by Sergey Shpectorov and by Paul Levy concentrated on involutions of affine buildings, respectively automorphisms of finite order of semisimple Lie algebras. The remaining three talks were more topologically in nature. Guy Rousseau presented his theory of microaffine buildings, hovels, and Kac–Moody groups over ultrametric fields on Thursday. Thursday's fourth talk was by Bertrand Rémy on Satake compactifications of buildings via Berkovich theory. The conference was concluded by Pierre-Emmanuel Caprace's report on aspects of the structure of locally compact groups. We are particularly pleased by the lively interaction between the participants during the long afternoon breaks (each morning's lectures finished at 11.30 a.m. while the afternoon sessions only started at 4.20 p.m.) and during the evenings.