Normal Cayley graphs of finite groups

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ◃A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG≅ℤ₄×ℤ₂ orG≅Q ₈×ℤ ₂ ^r (r⩾0) and that every finite group has a normal Cayley digraph, where Z_m is the cyclic group of orderm and Q₈ is the quaternion group of order 8. Project supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Doctorial Program Foundation of Institutions of Higher Education of China.     

Mappings of group shifts

A group shift is a proper closed shift-invariant subgroup ofG ^ℤ ₂ whereG is a finite group. We consider a class of group shifts in whichG is a finite field and show that mixing is a necessary and sufficient condition on such a group shift for all codes from it into another group shift to be affine and all codes from another group shift into it to be affine. As a corollary, it will follow forG=ℤ _p that two mixing group shifts are topologically conjugate if and only if they are equal.     

On the structure of the augmentation quotient group for some nonabelian 2-groups

Let G be a finite nonabelian group, ℤG its associated integral group ring, and Δ(G) its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups Q _n (G) = Δ ⁿ (G)/Δ ⁿ⁺¹(G) is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.     

Gorenstein Flat Covers and Gorenstein Cotorsion Modules Over Integral Group Rings

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module. Presented by A. Verschoren Mathematics Subject Classifications (2000) 20C05, 16E65.     

Schur multiplicators of infinite pro-<Emphasis Type="Italic">p</Emphasis>-groups with finite coclass

Let G be an infinite pro-p-group of finite coclass and let M(G) be its Schur multiplicator. For p > 2, we determine the isomorphism type of Hom(M(G), ℤ_p), where ℤ_p denotes the p-adic integers, and show that M(G) is infinite. For p = 2, we investigate the Schur multiplicators of the infinite pro-2-groups of small coclass and show that M(G) can be infinite, finite or even trivial.     

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G _ℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G _ℤ; that ifH is epimorphic inG then the action ofH onG/G _ℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G _ℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.     

Homology of free Lie powers and torsion in groups

LetG be a group that is given by a free presentationG=F/R, and let_γ4 R denote the fourth term of the lower central series of R. We show that ifG has no elements of order 2, then the torsion subgroup of the free central extensionF/[_γ4 R,F] can be identified with the homology groupR _γ6(G, ℤ/2ℤ). This is a consequence of our main result which refers to the homology ofG with coefficients in Lie powers of relation modules.     

Lifting representations of ℤ-groups

LetK be the kernel of an epimorphismG→ℤ, whereG is a finitely presented group. IfK has infinitely many subgroups of index 2,3 or 4, then it has uncountably many. Moreover, ifK is the commutator subgroup of a classical knot groupG, then any homomorphism fromK onto the symmetric groupS ₂ (resp. ℤ₃) lifts to a homomorphism ontoS ₃ (resp. alternating groupA ₄). Both authors partially supported by NSF grants DMS-0071004 and DMS-0304971.     

A simple map with no prime factors

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμ _S , i.e.,μ _S (A×B)=μ(A∩S ^;−n ;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕ _i=1 ^∞ ℤ₂, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.     

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

Rational and transcendental growth series for the higher Heisenberg groups

This paper considers growth series of 2-step nilpotent groups with infinite cyclic derived subgroup. Every such group G has a subgroup of finite index of the form H _n ×ℤ ^m , where H _n is the discrete Heisenberg group of length 2n+1. We call n the Heisenberg rank of G. We show that every group of this type has some finite generating set such that the corresponding growth series is rational. On the other hand, we prove that if G has Heisenberg rank n ≧ 2, then G possesses a finite generating set such that the corresponding growth series is a transcendental power series. Oblatum 1-III-1995 & 28-XII-1995     

On the Schur multipliers of finite p-groups of given coclass

In this paper we obtain bounds for the order and exponent of the Schur multiplier of a p-group of given coclass. These are further improved for p-groups of maximal class. In particular, we prove that if G is p-group of maximal class, then |H ₂(G, ℤ)| < |G| and expH ₂(G, ℤ) ≤ expG. The bound for the order can be improved asymptotically.     

The Littlewood-Gowers problem

The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions ƒ ∈ L ²(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space endowed with the norm , and generalize this to the approximate Fourier transform on Bohr sets. As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/3−ɛ; we improve this to ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ _A ‖ _A(ℤ/pℤ) ≪ log p.     

Universal lattices and property tau

We prove that the universal lattices – the groups G=SL_d(R) where R=ℤ[x₁,...,x_k], have property τ for d≥3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ-constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K₂ of some finite commutative rings and a relative property T of . Mathematics Subject Classification (2000) 20F69, 13M05, 19C20, 20G05, 20H05     

The bieri-strebel invariant and homological finiteness conditions for metabelian groups

A group Γ has type F P_n if a trivial ℤΓ-module ℤ has a projective resolution P:…P_n → … → P₁ → P₀ → ℤ in which ℤΓ-module P_n,…P₁, P₀ are finitely generated. Let the finitely generated group Γ be a split extension of the Abelian group M by an Abelian group Q, suppose M is torsion free, and assume Γ∈F P_m, m≥2. Then the invariant ∑ _c ^M is m-tame. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 194–218, March–April, 1997.     

Some zero-sum constants with weights

For an abelian group G, the Davenport constant D(G) is defined to be the smallest natural number k such that any sequence of k elements in G has a nonempty subsequence whose sum is zero (the identity element). Motivated by some recent developments around the notion of Davenport constant with weights, we study them in some basic cases. We also define a new combinatorial invariant related to (ℤ/nℤ) ^d , more in the spirit of some constants considered by Harborth and others and obtain its exact value in the case of (ℤ/nℤ)² where n is an odd integer.     

On the probabilistic min spanning tree Problem

We study a probabilistic optimization model for min spanning tree, where any vertex v _i of the input-graph G(V, E) has some presence probability p _i in the final instance G′ ⊂ G that will effectively be optimized. Suppose that when this “real” instance G′ becomes known, a spanning tree T, called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G′. The goal is to compute an anticipatory spanning tree of G such that, its modification for any G￠ í GG' \subseteq G is optimal for G′. This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively.