Uncountable Cofinalities of Permutation Groups

A sufficient criterion is found for certain permutation groupsG to have uncountable strong cofinality, that is, they cannotbe expressed as the union of a countable, ascending chain (H_i)_i     

On a Problem of P. Hall about Groups Isomorphic to all their Non-Trivial Normal Subgroups

A scheme of construction of infinite groups, other than simplegroups, free groups of infinite rank and the infinite cyclicgroup, which are isomorphic to all their non-trivial normalsubgroups is presented. Some results about the automorphismgroups of simple infinite groups are also obtained. In particular,it is proved that there is an infinite group G of any sufficientlylarge prime exponent p (or which is torsion-free) all of whoseproper subgroups are cyclic, and such that the groups Aut Gand Out G are isomorphic. The proofs use the technique of gradeddiagrams developed by A. Yu. Ol'shanskii. 1991 Mathematics SubjectClassification: 20F05, 20F06.     

Locally Finite Finitary Skew Linear Groups

Let V be a vector space over the division ring D of infinitedimension. We study locally finite, primitive groups G of finitarylinear automorphisms of V. We show that the derived group G'of G is infinite, simple, and lies in every non-trivial normalsubgroup of G, and that G' G Aut G'. Moreover if char D =0, then G is either the finitary symmetric group or the alternatinggroup on some infinite set. If D is commutative, that is, ifD is a field, then all these results are known and are the consequenceof the collective work of a number of people: in particular(in alphabetical order) V. V. Belyaev, J. I. Hall, F. Leinen,U. Meierfrankenfeld, R. E. Phillips, O. Puglisi, A. Radfordand quite probably others. 2000 Mathematics Subject Classification:20H25, 20H20, 20F50.     

On the Bergman Property for the Automorphism Groups of Relatively Free Groups

A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X^–1 of G, we have that G = X^k for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width.     

Automorphisms of Relatively Free Groups in the Variety N2A {wedge} AN2 {wedge} Nc

For positive integers n and c, with n 2, let G_{n, c} be a relativelyfree group of finite rank n in the variety N₂A AN₂ N_c. Itis shown that the subgroup of the automorphism group Aut(G_n,c) of G_{n, c} generated by the tame automorphisms and an explicitlydescribed finite set of IA-automorphisms of G_{n, c} has finiteindex in Aut(G_{n, c}). Furthermore, it is proved that there areno non-trivial elements of G_{n, c} fixed by every tame automorphismof G_{n, c}.     

On full groups of measure-preserving and ergodic transformations with uncountable cofinalities

The group of all measure-preserving permutations of the unitinterval and the full group of an ergodic transformation ofthe unit interval are shown to have uncountable cofinality andthe Bergman property. Here, a group G is said to have the Bergmanproperty if, for any generating subset E of G, some boundedpower of EE^–1{1} already covers G. This property arosein a recent interesting paper of Bergman, where it was derivedfor the infinite symmetric groups. We give a general sufficientcriterion for groups G to have the Bergman property. We showthat the criterion applies to a range of other groups, includingsufficiently transitive groups of measure-preserving, non-singular,or ergodic transformations of the reals; it also applies tolarge groups of homeomorphisms of the rationals, the irrationals,or the Cantor set.     

Symmetries of Surface Singularities

The study of reductive group actions on a normal surface singularityX is facilitated by the fact that the group Aut X of automorphismsof X has a maximal reductive algebraic subgroup G which containsevery reductive algebraic subgroup of Aut X up to conjugation.If X is not weighted homogeneous then this maximal group G isfinite (Scheja, Wiebe). It has been determined for cusp singularitiesby Wall. On the other hand, if X is weighted homogeneous butnot a cyclic quotient singularity then the connected componentG₁ of the unit coincides with the C* defining the weighted homogeneousstructure (Scheja, Wiebe, Wahl). Thus the main interest liesin the finite group G/G₁. Not much is known about G/G₁. Ganterhas given a bound on its order valid for Gorenstein singularitieswhich are not log-canonical. Aumann-Körber has determinedG/G₁ for all quotient singularities. We propose to study G/G₁ through the action of G on the minimalgood resolution of X. If X is weightedhomogeneous but not a cyclic quotient singularity, let E₀ bethe central curve of the exceptional divisor of . We show that the natural homomorphism GAut E₀ haskernel C* and finite image. In particular, this re-proves therest of Scheja, Wiebe and Wahl mentioned above. Moreover, itallows us to view G/G₁ as a subgroup of Aut E₀. For simple ellipticsingularities it equals (Z_bxZ_b)Aut₀ E₀ where –b is theself-intersection number of E₀, Z_bxZ_b is the group of b-torsionpoints of the elliptic curve E₀ acting by translations, andAut₀ E₀ is the group of automorphisms fixing the zero elementof E₀. If E₀ is rational then G/G₁ is the group of automorphismsof E₀ which permute the intersection points with the branchesof the exceptional divisor while preserving the Seifert invariantsof these branches. When there are exactly three branches weconclude that G/G₁ is isomorphic to the group of automorphismsof the weighted resolution graph. This applies to all non-cyclicquotient singularities as well as to triangle singularities.We also investigate whether the maximal reductive automorphismgroup is a direct product GG₁xG/G₁. This is the case, for instance,if the central curve E₀ is rational of even self-intersectionnumber or if X is Gorenstein such that its nowhere-zero 2-form has degree ±1. In the latter case there is a ‘natural’section G/G₁G of GG/G₁ given by the group of automorphisms inG which fix . For a simple elliptic singularity one has GG₁xG/G₁if and only if –E₀ · E₀ = 1.     

Latin Squares and the Hall-Paige Conjecture

The Hall–Paige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa trivial product. These two conditions are known to be equivalent,but the first direct, elementary proof that (i) implies (ii)is given here. It is also shown that the Hall–Paige conjecture impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a ‘double transversal’,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60.     

All Groups are Outer Automorphism Groups of Simple Groups

It is shown that each group is the outer automorphism groupof a simple group. Surprisingly, the proof is mainly based onthe theory of ordered or relational structures and their symmetrygroups. By a recent result of Droste and Shelah, any group isthe outer automorphism group Out (Aut T) of the automorphismgroup Aut T of a doubly homogeneous chain (T, ). However, AutT is never simple. Following recent investigations on automorphismgroups of circles, it is possible to turn (T, ) into a circleC such that Out (Aut T) Out (Aut C). The unavoidable normalsubgroups in Aut T evaporate in Aut C, which is now simple,and the result follows.     

Finite CI-Groups are Soluble

For a finite group G and a subset S of G with 1 S and S = S^–1,the Cayley graph Cay(G, S) is the graph with vertex set G suchthat {x, y} is an edge if and only if yx^–1 S. The groupG is called a CI-group if, for all subsets S and T of G\{1},Cay(G, S) Cay(G, T) if and only if S = T for some Aut(G).In this paper, for each prime p 1 (mod 4), a symmetric graph(p) is constructed from PSL(2, p) such that Aut (p) = Z₂ x PSL(2,p); it is then shown that A₅ is not a CI-group, and that allfinite CI-groups are soluble. 1991 Mathematics Subject Classification05C25.     

Invariants of Finite Reflection Groups and the Mean Value Problem for Polytopes

Let P be an n-dimensional polytope admitting a finite reflectiongroup G as its symmetry group. Consider the set H_P(k) of allcontinuous functions on Rⁿ satisfying the mean value propertywith respect to the k-skeleton P(k) of P, as well as the setH_G of all G-harmonic functions. Then a necessary and sufficientcondition for the equality H_P(k) = H_G is given in terms of adistinguished invariant basis, called the canonical invariantbasis, of G. 1991 Mathematics Subject Classification 20F55,52B15.     

Operator amenability of Fourier-Stieltjes algebras, II

We give an example of a non-compact, locally compact group Gsuch that its Fourier–Stieltjes algebra B (G) is operatoramenable. Furthermore, we characterize those G for which A *(G),the spine of B (G) as introduced by M. Ilie and N. Spronk, isoperator amenable and show that A *(G) is operator weakly amenablefor each G.     

The spine of a Fourier-Stieltjes algebra

We define the spine A *(G) of the Fourier–Stieltjes algebraB (G) of a locally compact group G. This algebra encodes informationabout much of the fine structure of B (G), particularly informationabout certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, thatof non-quotient locally precompact topologies on G. We computethe spine's spectrum G*, which admits a semi-group structure.We discuss homomorphisms from A *(G) to B (H) where H is anotherlocally compact group; and we show that A *(H) contains theimage of every completely bounded homomorphism from the Fourieralgebra A (H) of any amenable group G. We also show that A *(G)contains all of the idempotents in B (G). Finally, we computeexamples for vector groups, abelian lattices, minimally almostperiodic groups and the (ax + b)-group; and we explore the complexityof A *(G) for the discrete rational numbers and free groups.     

A Note on Two-Weight Inequalities for Maximal Functions and Singular Integrals

We deal with weighted inequalities of the type [formula] where: T is either the Hardy–Littlewood maximal operator or asingular integral operator; G is any measurable subset of Rⁿ; f is any measurable function, vanishing outside G, such thatTf is well-defined; v and w are weights, that is, nonnegativelocally integrable functions on G; p, q(1, ). 1991 Mathematics Subject Classification 42B20, 42B25.     

Uncountable Saturated Structures have the Small Index Property

We prove the following theorem. Let m be an uncountable saturatedstructure of cardinality = ^< and assume that G is a subgroupof Aut (m) whose index is less than or equal to . Then thereexists a subset A of cardinality strictly less than such thatevery automorphism of m leaving A pointwise fixed is in G.     

Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H_Ghas order at most n for every subgroup H of G (where H_G is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.     

Homomorphisms from Automorphism Groups of Free Groups

The automorphism group of a finitely generated free group isthe normal closure of a single element of order 2. If m <n, then a homomorphism Aut(F_n)Aut(F_m) can have image of cardinalityat most 2. More generally, this is true of homomorphisms fromAut(F_n) to any group that does not contain an isomorphic imageof the symmetric group S_n+1. Strong restrictions are also obtainedon maps to groups that do not contain a copy of W_n = (Z/2)ⁿ S_n, or of Z^n–1. These results place constraints on howAut(F_n) can act. For example, if n 3, any action of Aut(F_n)on the circle (by homeomorphisms) factors through det : Aut(F_n)Z₂.2000 Mathematics Subject Classification 20F65, 20F28 (primary).     

A Bound on the Presentation Rank of a Finite Group

Let G be a finite group, and let I_G be the augmentation idealof ZG. We denote by d(G) the minimum number of generators forthe group G, and by d(I_G) the minimum number of elements ofI_G needed to generate I_G as a G-module. The connection betweend(G) and d(I_G) is given by the following result due to Roggenkamp]14]: where pr(G) is a non-negative integer, called the presentationrank of G, whose definition comes from the study of relationmodules (see [4] for more details). 1991 Mathematics SubjectClassification 20D20.     

Projective Limits of Finite-Dimensional Lie Groups

For a topological group G we define N to be the set of all normalsubgroups modulo which G is a finite-dimensional Lie group.Call G a pro-Lie group if, firstly, G is complete, secondly,N is a filter basis, and thirdly, every identity neighborhoodof G contains some member of N. It is easy to see that everypro-Lie group G is a projective limit of the projective systemof all quotients of G modulo subgroups from N. The converseimplication emerges as a difficult proposition, but it is shownhere that any projective limit of finite-dimensional Lie groupsis a pro-Lie group. It is also shown that a closed subgroupof a pro-Lie group is a pro-Lie group, and that for any closednormal subgroup N of a pro-Lie group G, for any one parametersubgroup Y : R G/N there is a one parameter subgroup X : R G such that X(t) N = Y(t) for any real number t. The categoryof all pro-Lie groups and continuous group homomorphisms betweenthem is closed under the formation of all limits in the categoryof topological groups and the Lie algebra functor on the categoryof pro-Lie groups preserves all limits and quotients. 2000 MathematicsSubject Classification 22E65, 22D05, 22E20, 22A05, 54B35.     

On the Exceptional Principal Bundles Over a Surface

Let G be a simple simply connected complex Lie group. Some criteriaare given for the nonexistence of exceptional principal G-bundlesover a complex projective surface. As an application, it isshown that there are no exceptional G-bundles over a surfacewhose arithmetic genus is zero or one. It is also shown thatthere are no stable exceptional G-bundles over an abelian surface.2000 Mathematics Subject Classification 32L20, 14J60.