Blocks with cyclic defect of Hecke orders of Coxeter groups

Let B\cal B be a p-block of cyclic defect of a Hecke order over the complete ring \Bbb Z[q] _{áq-1,p ?}\Bbb {Z}[q] _{\langle q-1,p \rangle}; i.e. modulo áq-1 ?\langle q-1 \rangle it is a p-block B of cyclic defect of the underlying Coxeter group G. Then B\cal B is a tree order over \Bbb Z[q]_{áq-1, p ?}\Bbb {Z}[q]_{\langle q-1, p \rangle } to the Brauer tree of B. Moreover, in case B\cal B is the principal block of the Hecke order of the symmetric group S(p) on p elements, then B\cal B can be described explicitly. In this case a complete set of non-isomorphic indecomposable Cohen-Macaulay B\cal B-modules is given.     

Kazhdan's property (T), L^2-spectrum and isoperimetric inequalities for locally symmetric spaces

Let V = G\G/KV =\Gamma\backslash G/K be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups G ì G\Gamma\subset G in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).     

Group-invariant Percolation on Graphs

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group.¶ We show that G is amenable if for all $ \alpha < 1 $ \alpha < 1 , there is a G-invariant site percolation process w \omega on X with $ {\bf P} [x \in \omega] > \alpha $ {\bf P} [x \in \omega] > \alpha for all vertices x and with no infinite components. When G is not amenable, a threshold $ \alpha < 1 $ \alpha < 1 appears. An inequality for the threshold in terms of the isoperimetric constant is obtained, extending an inequality of Häggström for regular trees.¶ If G acts transitively on X, we show that G is unimodular if the expected degree is at least 2 in any G-invariant bond percolation on X with all components infinite.¶ The investigation of dependent percolation also yields some results on automorphism groups of graphs that do not involve percolation.     

Orbits of permutation groups on the power set

Let G be a permutation group on a finite set W\Omega . If G does not involve A_n for n \geqq 5 n \geqq 5 , then there exist two disjoint subsets of W\Omega such that no Sylow subgroup of G stabilizes both and four disjoint subsets of W\Omega whose stabilizers in G intersect trivially.     

Hall polynomials and the fixed space of an automorphism of a finite abelian p-group

Recent work on Hall polynomials is used to study the fixed space of a random automorphism of a finite abelian p-group. An expression is found for the chance that an automorphism of an abelian p-group of type l\lambda fixes only the identity. A formula is obtained for the chance that a given subgroup H of type n\nu is the fixed space of an automorphism of an abelian p-group of type (k^l). There results generalize work of Rudvalis and Shinoda on the fixed space of an element of GL (n, q).     

Complex finitary simple Lie algebras

An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over an algebraically closed field of zero characteristic. It is shown that any such algebra is isomorphic to one of the following¶ (1) a special transvection algebra \frak t(V,P)\frak t(V,\mit\Pi );¶ (2) a finitary orthogonal algebra \frak fso (V,q)\frak {fso} (V,q); ¶ (3) a finitary symplectic algebra \frak fsp (V,s)\frak {fsp} (V,s).¶Here V is an infinite dimensional K-space; q (respectively, s) is a symmetric (respectively, skew-symmetric) nondegenerate bilinear form on V; and P\Pi is a subspace of the dual V^* whose annihilator in V is trivial: 0={v ? V | Pv=0}0=\{{v}\in V\mid \Pi {v}=0\}.     

Idempotents in a complex group algebra, projective modules, and the von Neumann algebra

The complex group algebra \Bbb CG{\Bbb C}G of a countable group G can be imbedded in the von Neumann algebra NG of G. If G is torsion-free, and if P is a finitely generated projective module over \Bbb CG{\Bbb C}G it is proved that the central-valued trace of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P, i.e. of an idempotent \Bbb CG{\Bbb C}G-matrix A defining P is equal to the canonical trace k(P)\kappa (P) times identity I. It follows that k(P)\kappa (P) characterizes the isomorphism type of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P.¶If k(P)\kappa (P) is an integer, e.g., if the weak Bass conjecture holds for G then NG?_{\Bbb C G}PNG\otimes _{{\Bbb C} G}P is free. It is also shown that for certain classes of groups geometric arguments can be used to prove the Bass conjecture.     

A combinatorial condition on a certain variety of groups

Let n be an integer and B_n \mathcal B_n be the variety defined by the law [xⁿ,y][x,yⁿ]^-1 = 1.¶ Let B_n^* \mathcal B_n^* be the class of groups in which for any infinite subsets X, Y there exist x ? X x \in X and y ? Y y \in Y such that [xⁿ,y][x,yⁿ]^-1 = 1. For $ n \in {\pm 2, 3\} $ n \in {\pm 2, 3\} we prove that¶ B_n^* = B_n èF \mathcal B_n^* = \mathcal B_n \cup \mathcal F , F \mathcal F being the class of finite groups. Also for $ n \in {- 3, 4\} $ n \in {- 3, 4\} and an infinite group G which has finitely many elements of order 2 or 3 we prove that G ? B_n^* G \in \mathcal B_n^* if and only if G ? B_n G \in \mathcal B_n .     

The character values of the irreducible constituents of a transitive permutation representation

In this article we determine the irreducible ordinary characters c_r \chi_r of a finite group G occurring in a transitive permutation representation (1_M )^G of a given subgroup M of G, and their multiplicities m_r = ((1_M)^G, c_r) 1 0 m_r = ((1_{M})^G, \chi_r) \neq 0 by means of a new explicit formula calculating the coefficients a_rk of the central idempotents e_r = ?_k=1^d a_rk D_k e_r = \sum\limits_{k=1}^{d} a_{rk} D_k in the intersection algebra B \cal B of (1_M )^G generated by the intersection matrices D_k corresponding to the double coset decomposition G = è_k=1^d Mx_k M G = \bigcup\limits_{k=1}^{d} Mx_{k} M .¶Furthermore, an explicit formula is given for the calculation of the character values c_r(x) \chi_{r}(x) of each element x ? G x \in G . Using this character formula we obtain a new practical algorithm for the calculation of a substantial part of the character table of G.     

On groups acting freely on a tree

Suppose we are given a group G\mit\Gamma and a tree X on which G\mit\Gamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers a_d: = # {w ? vertX : w=gv, g ? G , d(v₀,w)=d }a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \} if d? ￥d\rightarrow \infty , where v, v_o are some fixed vertices in X.¶ In this paper we consider the case where G\mit\Gamma is a finitely generated group acting freely on a tree X. The growth function ?a_d x^d\textstyle\sum\limits a_d x^d is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where G\mit\Gamma is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.     

A characterization of infinite 3-abelian groups

In this note we prove that every infinite group G is 3-abelian (i.e. (ab)³ = a³b³ for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ? Xx\in X and y ? Yy\in Y such that (xy)³ = x³y³.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let t: D ?D￠\tau: {\cal D} \rightarrow{\cal D}^\prime be an equivariant holomorphic map of symmetric domains associated to a homomorphism r: \Bbb G ?\Bbb G￠{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime of semisimple algebraic groups defined over \Bbb Q{\Bbb Q} . If G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q}) and G￠ ì \Bbb G￠(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q}) are torsion-free arithmetic subgroups with r (G) ì G￠{\bf\rho} (\Gamma) \subset \Gamma^\prime , the map G\D ?G￠\D￠\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime of arithmetic varieties and the rationality of D{\cal D} and D￠{\cal D}^\prime as well as the commensurability groups of s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C}) determines a conjugate equivariant holomorphic map t^s: D^s ?D^￠s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma} of f^s: (G\D)^s ?(G￠\D￠)^s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma of . We prove that is rational if is rational.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

On standard Auslander-Reiten sequences for finite groups

Let G be a finite group and O{\cal O} a complete discrete valuation ring of characteristic zero with maximal ideal (p)(\pi ) and residue field k = O/(p)k = {\cal O}/(\pi ) of characteristic p > 0. Let S be a simple kG-module and Q_S a projective O G{\cal O} G-lattice such that Q_S / pQ_SQ_S / \pi Q_S is a projective cover of S. We show that if S is liftable and Q_S belongs to a block of O G{\cal O} G of infinite representation type, then the standard Auslander-Reiten sequence terminating in W^-1S\Omega ^{-1}S is a direct summand of the short exact sequence obtained from some Auslander-Reiten sequence of OG{\cal O}G-lattices by reducing each term mod (p)(\pi ).     

On p-chief factors and extensions of $ \mathbb{K} $ G-modules

The subject of this paper is the relationship between the set of chief factors of a finite group G and extensions of an irreducible \mathbbK \mathbb{K} G-module U ( \mathbbK \mathbb{K} a field). Let H / L be a p-chief factor of G. We prove that, if H / L is complemented in a vertex of U, then there is a short exact sequence of Ext-functors for the module U and any \mathbbK \mathbb{K} G-module V. In some special cases, we prove the converse, which is false in general. We also consider the intersection of the centralizers of all the extensions of U by an irreducible module and provide new bounds for this group.     

The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups

Let G be a finite group. Two subgroups H and K of G are said to permute if áH,K? = HK = KH\langle H,K\rangle = HK = KH. A subgroup H of G is S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we investigate the influence of S-quasinormality of some subgroups of prime power order of a finite group on its supersolvability.     

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability property of a unitary representation p \pi of a group G (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) , where \Bbb S_H {\Bbb S}_{\cal H} is the unit sphere the Hilbert space of representation. We prove that p \pi is amenable if and only if either p \pi contains a finite-dimensional subrepresentation or the maximal uniform compactification of (\Bbb S_p ({\Bbb S}_{\pi} has a G-fixed point. Equivalently, the latter means that the G-space (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) has the concentration property: every finite cover of the sphere \Bbb S_p {\Bbb S}_{\pi} contains a set A such that for every e > 0 \epsilon > 0 the e \epsilon -neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of p \pi is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \Bbb S_p {\Bbb S}_{\pi} . As another corollary, a locally compact group G is amenable if and only if for every strongly continuous unitary representation of G in an infinite-dimensional Hilbert space H {\cal H} the system (\Bbb S_H, G) ({\Bbb S}_{\cal H}, G) has the property of concentration.     

Efficient Testing of Large Graphs

Let P be a property of graphs. An e\epsilon -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than en²\epsilon n^2 edges to make it satisfy P. The property P is called testable, if for every e\epsilon there exists an e\epsilon -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain individual graph properties, like k-colorability, admit an e\epsilon -test. In this paper we make a first step towards a complete logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type ``"$\forall \exists ' are always testable, while we show that some properties containing this alternation are not.     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.