A rigidity theorem for group extensions

We consider the class of discrete groups which arise as fundamental groups of iterated surface fibrations; that is, of complexes obtained from a sequence of fibrations in which all bases and the initial fibre are hyperbolic surfaces. Group theoretically, this corresponds to studying the class of iterated extensions of hyperbolic surface groups. In [4], for the case of a single extension we conjectured and partially established that no group can arise from more than a finite number of such extensions. Here we show that the result holds in complete generality. As remarked in [4], the result has a strong affinity with the rigidity theorems of Parshin [7] and Arakelov [1] for fibred (complex) algebraic surfaces.     

Self-dual normal integral bases for infinite unramified extensions

We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.     

Pattern analysis of the genetic code

The genetic code is examined in a new and systematic fashion: we consider the code as mapping of one finite set (the 64 codons) to another (the 20 amino acids). Given a class of mappings simpler than the actual code, we ask which mappings best approximate it. This leads to an analysis of the effects of ambiguities (codon degeneracy) in one or two positions. With the 0–1 metric (counting the amino acids as equal or not equal), the codon third base degeneracy is apparent, but the first and second positions are indistinguishable; with the integrated amino acid “distance” metric compiled by Sneath (J. Theoret. Biol. 12 (1966), 157–195), the analysis ranks the information content of the three codon positions as follows: 2nd > 1st > 3rd. We discuss possible further applications of this approach to patterns in the genetic code and other codes.     

Globally irreducible representations of SL3(q) and SU3(q)

The notion ofglobally irreducible representations of finite groups was introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered by N. Elkies and T. Shioda using Mordell-Weil lattices of elliptic curves. In this paper we classify all globally irreducible representations coming from projective complex representations of the finite simple groups PSL₃(q) and PSU₃(q). The main result is that these representations are essentially those discovered by Gross.     

Distance-transitive representations of the sporadic groups

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free.

In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.     

Coleman automorphisms of finite groups and their minimal normal subgroups

In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic p-group. Finally, we note that class-preserving Coleman automorphisms of p-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge, where p is a prime number.     

关于Brauer特征标扩张的一点注记

研究有限群特征标可扩张的情况是有限群表示论领域中一个有意义的问题.设G为有限群,用Irr（G）表示G的所有不可约复特征标构成的集合.设N（？）G,θ∈Irr（N）且θ是G-不变.如果（｜G/N｜,o（θ）θ（1））=1,则[1]中的推论8.16说明了此时υ到G有唯一的扩张χ,且o（χ）=o（θ）.此结论启发了我们可以从特征标的行列式阶的角度来思考特征标扩张的情形.本文将利用有限群Brauer特征标的行列式阶,着重考虑Brauer特征标的可扩张情形.另外我们也得到了一个关于Brauer特征标次数的结论.     

Unitary Representations of Oligomorphic Groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S ^??, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand?CRaikov theorem holds for topological subgroups of S ^??: for all such groups, continuous irreducible representations separate points in the group.     

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

群代数Ore扩张的量子群

本文讨论由Beattie通过群代数的Ore扩张所构造的量子群的中心和Yang－Baxter模范畴．此外,完全刻画了这类量子群的不可约表示．     

Abelian Extensions and Solvable Loops

Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations between nilpotence and solvability for loops and the associated multiplication groups and inner mapping groups.     

Cayley Digraphs of Finite Simple Groups with Small Out-Valency

Abstract

For a group G and a subset S of G which does not contain the identity of G, the Cayley digraph Cay(G, S) is called normal if R(G) is normal in Aut(Γ). In this paper, we investigate the normality of Cayley digraphs of finite simple groups with out-valency 2 and 3. We give several sufficient conditions for such Cayley digraphs to be normal. By using this result, we consider the digraphical regular representations of finite simple groups.     

On n ‐contractive fuzzy logics

It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL‐chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL‐chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Moduli problem and points on some twisted Shimura varieties of PEL type

In this article we describe the moduli problem of a “twist” of some simple Shimura varieties of PEL type that appear in Kottwitz's papers [R. Kottwitz, Shimura varieties and λ-adic representations, in: Automorphic Forms, Shimura Varieties and L-Functions, part 1, in: Perspect. Math., vol. 10, Academic Press, San Diego, CA, 1990, pp. 161-209; R. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (2) (1992) 373-444] and [R. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992) 653-665] and then, using the moduli problem, we compute the cardinality of the set of points over finite fields of the twisted Shimura varieties. Using this result, we compute the zeta function of the twisted varieties. The twist of the Shimura varieties is done by a mod q representation of the absolute Galois group of the reflex field of the Shimura varieties.     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

On Generic Extensions of Representations of a Dynkin Quiver of Type D

The notion of generic extensions of representations of a Dynkin quiver plays a big role in the study of the structure of the corresponding quantum group. In this paper, we describe the generic extensions of a simple representation by any representation and that of any representation by a simple representation of a Dynkin quiver Q of type D.     

Finite groups with a five-component prime graph

Some classification is obtained for the finite groups whose prime graph has five connected components. In particular, we show that such a group is simple. Some related results are stated about representations of simple groups.     

Extensions of Finite Quantum Groups by Finite Groups

We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in [AG] as quantum subgroups of quantum groups at roots of 1. Finally, we apply the first general result to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups. Partially supported by CONICET, ANPCyT, Secyt (UNC) and Ministerio de Ciencia y Tecnología de la Provincia de Córdoba.     

Characterization of Finite Simple Nonabelian Groups via Twisted Sets

We prove that if the class of the groups generated by the same twisted subset (up to isomorphism) contains a finite simple nonabelian group then the groups of the class are finite perfect central extensions of the simple group.     

Invariant properties of representations under cleft extensions

The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a relation between the cleft extension and the crossed product which is the approach we depend upon. Then, by making use of them, we prove that over an algebraically closed field k, for a finite dimensional Hopf algebra H which is semisimple as well as its dual H*, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension . In the other part, we still show that over an arbitrary field k, the Nakayama property of a k-algebra is also an invariant property under an H -cleft extension when the radical of the algebra is H-stable.