Zusammenfassung.- Introduction.- Fundamentals.- General results and methods.- Applications.- Bibliopraphy.- Index.- List of tables.

Blocks of Finite Groups and Their Invariants

A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G-designs. This includes references to all the necessary designs and constructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 373-410, 2008

A survey on the existence of G-Designs

Publisher Summary A balanced incomplete block design (BIBD) is a pair ( V,B ) where V is a ν set and B is a collection of b k- subsets of V called “blocks” such that each element of V is contained in exactly r blocks, and any 2-subset of V is contained in exactly X blocks. The numbers ν, b, r, k, λ are parameters of the BIBD. A BIBD ( V,B ) is resolvable if there exists a partition R of its set of blocks B into subsets called “parallel classes,” each of which in turn partitions the set V ; R is called a “resolution.” Given a symmetric BIBD (one with u = b , r = k ) , the residual design by deleting all elements of one block and the derived design by deleting all elements of the complement of one block can be obtained.

Tables of Parameters of BIBDs with r⩽41 including Existence, Enumeration, and Resolvability Results

Introduction Higher limits over orbit categories Reduction to simple groups A relative version of $\Lambda$-functors Subgroups which contribute to higher limits Alternating groups Groups of Lie type in characteristic two Classical groups of Lie type in odd characteristic Exceptional groups of Lie type in odd characteristic Sproadic groups Computations of $\textrm{lim}^1(\mathcal{Z}_G)$ Bibliography.

Equivalences of Classifying Spaces Completed at the Prime Two

It is indeed unfortunate that Richard Brauer did not live to see the complete classification of the finite simple groups. He had devoted the past thirty years largely to their study and it is difficult to overestimate the impact he made on the subject. Early on, he realized the intimate relationship between the structure of a group and the centralizers of its involutions (elements of order 2). He established both qualitative and quantitative connections. As an example of the first, he showed by an elementary argument that there are at most a finite number of simple groups with a specified centralizer of an involution [30]; and of the second, he proved that if the centralizer of an involution in a simple group G is isomorphic to the general linear group GL(2, q) over the finite field with q elements, q odd, then either G is isomorphic to the three-dimensional projective special linear group L3(q), 2 or else q = 3 and G is isomorphic to the smallest Mathieu group Mn of order 8 • 9 • 10 • 11 [27], [28]. This last result, which Brauer announced in his address at the International Congress of Mathematicians in Amsterdam in 1954, represented the starting point for the classification of simple groups in terms of the structure of the centralizers of their involutions. Moreover, it foreshadowed the basic fact that conclusions of general classification theorems would necessarily include sporadic simple groups as exceptional cases (Mn being the sporadic group of least order). The methods which Brauer used were almost entirely representationtheoretic and character-theoretic. In the middle 1930s he introduced and developed the concept of modular characters of a finite group. He soon realized the power of these ideas, which played an instrumental role in his proof of the Artin conjecture on Ê-series in algebraic number fields. Likewise he saw that these techniques provided a powerful tool for investigating simple groups. From the middle 1940s until his death, Brauer systematically devel-

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The classification of finite simple groups I. Simple groups and local analysis

Publisher Summary This chapter discusses coset enumeration in groups and constructions of symmetric designs. It presents a group in terms of generators and relations so that each point or block stabilizer, can be expressed with the same generators. The Coxeter–Todd coset enumeration method with respect to the subgroup gives the number of cosets and also gives the permutation representation of group with respect to the (right) cosets of the subgroup. The corresponding programs have been made by Hrabe De Angelis for each stabilizer subgroup. To construct the design means only to put together all these permutation representations according to the orbit structure matrix. In JANKO–TRAN a symmetric design was constructed whose full automorphism group is discussed.

Coset Enumeration in Groups and Constructions of Symmetric Designs

A new finite simple group of order 86 · 775 · 571 · 046 · 077 · 562 · 880 which possesses M24 and the full covering group of M22 as subgroups

Construction of a new symmetric block design for (78, 22, 6) with the help of tactical decompositions

The Existence of a Bush-type Hadamard Matrix of Order 36 and Two New Infinite Classes of Symmetric Designs

A symmetric 2-(324, 153, 72) design is constructed that admits a tactical decomposition into 18 point and block classes of size 18 such that every point is in either 0 or 9 blocks from a given block class, and every block contains either 0 or 9 points from a given point class. This design is self-dual and yields a symmetric Hadamard matrix of order 324 of Bush type, being the first known example of a symmetric Bush-type Hadamard matrix of order 4n^2 for n > 1 odd. Equivalently, the design yields a strongly regular graph with parameters v=324, k=153, \lambda=\mu=72 that admits a spread of cocliques of size 18. The Bush-type Hadamard matrix of order 324 leads to two new infinite classes of symmetric designs with parameters

v=324(289^m+289^{m-1}+\cdots+289+1), \quad k=153(289)^m, \quad \lambda=72(289)^m,

and

v=324(361^m+361^{m-1}+\cdots+361+1), \quad k=171(361)^m, \quad \lambda=90(361)^m,

where m is an arbitrary positive integer.

Zvonimir Janko

Papers

Coset Enumeration in Groups and Constructions of Symmetric Designs

A new finite simple group of order 86 · 775 · 571 · 046 · 077 · 562 · 880 which possesses M24 and the full covering group of M22 as subgroups

Construction of a new symmetric block design for (78, 22, 6) with the help of tactical decompositions

The Existence of a Bush-type Hadamard Matrix of Order 36 and Two New Infinite Classes of Symmetric Designs

The Existence of a Bush-Type Hadamard Matrix of Order 324 and Two New Infinite Classes of Symmetric Designs