This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4 For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras

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The Book of Involutions

Let G " be a non-directed graph having n vertices, without parallel edges and slings. Let the vertices of Gn be denoted by F 1 ,. . ., Pn. Let v(P j) denote the valency of the point P i and put (0. 1) V(G,) = max v(Pj). 1ninn Let E(G.) denote the number of edges of Gn. Let H d (n, k) denote the set of all graphs Gn for which V (G n) = k and the diameter D (Gn) of which is-d, In the present paper we shall investigate the quantity (0 .2) Thus we want to determine the minimal number N such that there exists a graph having n vertices, N edges and diameter-d and the maximum of the valencies of the vertices of the graph is equal to k. To help the understanding of the problem let us consider the following interpretation. Let be given in a country n airports ; suppose we want to plan a network of direct flights between these airports so that the maximal number of airports to which a given airport can be connected by a direct flight should be equal to k (i .e. the maximum of the capacities of the airports is prescribed), further it should be possible to fly from every airport to any other by changing the plane at most d-1 times ; what is the minimal number of flights by which such a plan can be realized? For instance, if n = 7, k = 3, d= 2 we have F2 (7, 3) = 9 and the extremal graph is shown by Fig. 1. The problem of determining Fd (n, k) has been proposed and discussed recently by two of the authors (see [1]). In § 1 we give a short summary of the results of the paper [1], while in § 2 and 3 we give some new results which go beyond those of [1]. Incidentally we solve a long-standing problem about the maximal number of edges of a graph not containing a cycle of length 4. In § 4 we mention some unsolved problems. Let us mention that our problem can be formulated also in terms of 0-1 matrices as follows : Let M=(a il) be a symmetrical n by n zero-one matrix such 2

On a problem of graph theory

Preface.- Introduction.- Glossary of Notation.- I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Basic Concepts.- 4 Terms, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Terms and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- III Congruences.- 1 Congruence Spreading.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- IV Lattice Constructions.- 1 Adding an Element.- 2 Gluing.- 3 Chopped Lattices.- 4 Constructing Lattices with Given Congruence Lattices.- 5 Boolean Triples.- V Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- VI Varieties of Lattices.- 1 Characterizations of Varieties 397.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- VII Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Afterword.- Bibliography.

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Lattice Theory: Foundation

Homology and central series of groups

Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe

Linear algebra and projective geometry

Engelsche Elemente Noetherscher Gruppen

The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to -g in G. The significance of elements of prime power index is best recognized once one remembers Wielandt's Theorem that elements whose order and index are powers of the same prime p are contained in a normal subgroup of order a power of p and Burnside's Theorem asserting the absence of elements of prime power index, not 1, in simple groups. From Burnside's Theorem one deduces easily that a group without proper characteristic subgroups contains an element, not 1, whose index is a power of a prime if and only if this group is abelian. In this result it suffices to assume the absence of proper fully invariant subgroups, since we can prove [in ?2] the rather surprising result that a [finite] group does not possess proper fully invariant subgroups if and only if it does not possess proper characteristic subgroups. A deeper insight will be gained if we consider groups which contain "many" elements of prime power index. We show [in ?5 ] that the elements of order a power of p form a direct factor if, and only if, their indices are powers of p too; and nilpotency is naturally equivalent to the requirement that this property holds for every prime p. More difficult is the determination of groups with the property that every element of prime power order has also prime power index [?3]. It follows from Burnside's Theorem that such groups are soluble; and it is clear that a group has this property if it is the direct product of groups of relatively prime orders which are either p-groups or else have orders divisible by only two different primes and furthermore have abelian Sylow subgroups. But we are able to show conversely that every group with the property under consideration may be represented in the fashion indicated. In ?5 we study the so-called hypercenter. This characteristic subgroup has been defined in various ways: as the terminal member of the ascending central chain or as the smallest normal subgroup modulo which the center is 1. We may add here such further characterizations as the intersection of all the normalizers of all the Sylow subgroups or as the intersection of all the maximal nilpotent subgroups; and the connection with the index problem is obtained by showing that a normal subgroup is part of the hypercenter if, and only if, its elements of order a power of p have also index a powrer of p. Notation. All the groups under consideration will be finite. An element [group] is termed primary, if its order is a prime power;

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Group elements of prime power index

It is a well known theorem that an abelian group G satisfying G = nG for every positive integer n is a direct summand of every abelian group H which contains G as a subgroup. It is the object of this note to generalize this theorem to abelian groups admitting a ring of operators, and to show that the corresponding conditions are not only sufficient but are at the same time necessary. Finally we show that every abelian group admitting a ring of operators may be imbedded in a group with the above mentioned properties; and it is possible to choose this \"completion\" of the given group in such a way that it is isomorphic to a subgroup of every other completion. Our investigation is concerned with abelian groups admitting a ring of operators. A ring R is an abelian group with regard to addition, its multiplication is associative, and the two operations are connected by the distributive laws. As the multiplication in R need not be commutative, we ought to distinguish left-, rightand two-sided ideals. Since, however, only left-ideals will occur in the future, we may use the term \"ideals\" without fear of confusion. Thus an ideal in R is a non-vacuous set M of elements in R with the property : If m', m\" are elements in M, and if r', r \" are elements in R, then r'm'±r\"m\" is an element in M. An abelian group G whose composition is written as addition is said to admit the elements in the ring R as operators (or shorter: G is an abelian group over R), if with every element r in R and g in G is connected their uniquely determined product rg so that this product is an element in G and so that this multiplication satisfies the associative and distributive laws. If G is an abelian group over R, then its subgroups M are characterized by the same property as the ideals MinR. We assume finally the existence of an element 1 in R so that g = lg for every g in G and r • 1 = 1 • r = r for every r in R. If x is any element in the abelian group G over Ry then its order N(x) consists of all the elements r in R which satisfy nc = 0. One verifies that every order N(x) is an ideal in 2?, and that N(x) =R if, and only if, x = 0. If M is an ideal in R, and if x is an element in G, then a subgroup

https://projecteuclid.org/download/pdf_1/euclid.bams/1183503234

Reinhold Baer

Papers

Linear algebra and projective geometry

Engelsche Elemente Noetherscher Gruppen

Group elements of prime power index

Abelian groups that are direct summands of every containing abelian group

Partitionen endlicher Gruppen