Barely Transitive and Heineken Mohamed Groups

A negative answer to the Kuro–ernikov Question 21 in [7],whether a group satisfying the normalizer condition is hypercentral,was given by Heineken and Mohamed in 1968 [6]. They constructedgroups G satisfying: (i) G is a locally finite p-group for a prime p, (ii) G/G'C_p and G' is countable elementary abelian, (iii) every proper subgroup of G is subnormal and nilpotent, (iv) Z(G)={1}, (v) the set of normal subgroups of G contained in G' is linearlyordered by set inclusion, see [3, p. 334], (vi) KG' is a proper subgroup in G for every proper subgroupK of G, see [6, Lemma 1(a)].     

Derivations with a Hereditary Domain

We investigate the closability of those derivations D definedon a (non necessarily closed) subalgebra B of a complex Banachalgebra A for which the conditions BABB and dim[B^kRad(A)]< hold for some kN, where Rad(A) stands for the Jacobson radicalof A. In this situation we show that the separating subspaceY(D) for D satisfies the property B[BY(D)]BRad(A). Furthermore, we demonstrate several specially relevant situationsin which we get a ‘closability property’ which ismore precise than the former one.     

Restricted Set Addition in Groups I: The Classical Setting

The existing results are surveyed and several new results areproved for the cardinality of the restricted doubling 2^A ={a' + a':a',a' A,a' a'}, where A G is a subset of theset of elements of an (additively written) group G. In particular,known estimates for G = Z and G = Z/pZ are improved and a first-of-a-kindgeneral estimate valid for arbitrary G is given.     

Morita Equivalent Blocks in Non-Normal Subgroups and p-Radical Blocks in Finite Groups

Let O be a complete discrete valuation ring with unique maximalideal J(O), let K be its quotient field of characteristic 0,and let k be its residue field O/J(O) of prime characteristicp. We fix a finite group G, and we assume that K is big enoughfor G, that is, K contains all the |G|-th roots of unity, where|G| is the order of G. In particular, K and k are both splittingfields for all subgroups of G. Suppose that H is an arbitrarysubgroup of G. Consider blocks (block ideals) A and B of thegroup algebras RG and RH, respectively, where R{O, k}. We considerthe following question: when are A and B Morita equivalent?Actually, we deal with ‘naturally Morita equivalent blocksA and B’, which means that A is isomorphic to a full matrixalgebra of B, as studied by B. Külshammer. However, Külshammerassumes that H is normal in G, and we do not make this assumption,so we get generalisations of the results of Külshammer.Moreover, in the case H is normal in G, we get the same resultsas Külshammer; however, he uses the results of E. C. Dade,and we do not.     

The Glauberman Correspondence and Subgroups of Operator Groups: a Counterexample

Let G and A be finite groups with coprime orders, and supposethat A acts on G by automorphisms. Let (G, A):Irr_A(G)Irr(C_G(A))be the Glauberman–Isaacs correspondence. Let B A andIrr_A(G). We exhibit a counterexample to the conjecture that(G, A) is an irreducible constituent of the restriction of (G,B) to C_G(A). 1991 Mathematics Subject Classification 20C15.     

On the Small Squaring and Commutativity

Let G be a group and let k > 2 be an integer, such that (k²– 3)(k – 1) < |G|/15 if G is finite. Supposethat the condition |A²| k(k + 1)/2 + (k – 3)/2 is satisfiedby every it-element subset A G. Then G is abelian. The proofuses the structure of quasi-invariant sets.     

Normal Subgroups of Groups Which Split Over The Infinite Cyclic Group

Let G be either a free product with amalgamation A*_CB or anHNN group A*_C, where all normal subgroups of C are finitelygenerated. Suppose that both A and B have no non-trivial finitelygenerated normal subgroups of infinite indices. We show thatif G contains a finitely generated normal subgroup N which intersectsA or B non-trivially but is not contained in C, then the indexof N in G is finite. 1991 Mathematics Subject Classification20E06.     

Actions of Commutative Hopf Algebras

We show that actions of finite-dimensional semisimple commutativeHopf algebras H on H-module algebras A are essentially group-gradings.Moreover we show that the centralizer of H in the smash productA # H equals A^H H. Using these we invoke results about groupgraded algebras and results about centralizers of separablesubalgebras to give connections between the ideal structureof A, A^H and A # H. Examples of the above occur naturally when one considers: (1) finite abelian groups G of automorphisms of an algebra Awith | G |^–1 A; (2) G-graded algebras, for finite groups G; (3) finite-dimensional restricted Lie algebras L, with semisimplerestricted enveloping algebra u(L), acting as derivations onan algebra A.     

Connected Lie Groups that Act Freely on a Product of Linear Spheres

In this note we prove that a compact connected Lie group G admitsa free action on some product of linear spheres if and onlyif it is isomorphic to (T^k x SU(2)^l)/Z for some k and l andfor some central elementary abelian 2-subgroup Z with Z SU(2)M^l= 1.     

On Semidirect Products and the Arithmetic Lifting Property

Let G be a finite group and let K be a hilbertian field. Manyfinite groups have been shown to satisfy the arithmetic liftingproperty over K, that is, every G-Galois extension of K arisesas a specialization of a geometric branched covering of theprojective line defined over K. The paper explores the situationwhen a semidirect product of two groups has this property. Inparticular, it shows that if H is a group that satisfies thearithmetic lifting property over K and A is a finite cyclicgroup then G = A H also satisfies the arithmetic lifting propertyassuming the orders of H and A are relatively prime and thecharacteristic of K does not divide the order of A. In thiscase, an arithmetic lifting for any AH-Galois extension of Kis explicitly constructed and the existence of the arithmeticlifting for any G-Galois extension is deduced. It is also shownthat if A is any abelian group, and H is the group with thearithmetic lifting property then AH satisfies the property aswell, with some assumptions on the ground field K. In the constructionproperties of Hilbert sets in hilbertian fields and spectralsequences in étale cohomology are used.     

Isomorphisms of Cayley graphs of a free Abelian group

A group G is called a CI-group provided that the existence of some automorphism σ ∈ Aut(G) such that σ(A) = B follows from an isomorphism Cay(G, A) ? = Cay (G, B) between Cayley graphs, where A and B are two systems of generators for G. We prove that every finitely generated abelian group is a CI-group.     

On the product of two Chernikov subgroups

It is shown that if a group G = AB is the product of two subgroups A and B, each of which has an abelian subgroup of index at most 2 satisfying the minimum condition and such that one of the subgroups A or B is of dihedral type, then G is abelian-by-finite with minimum condition.     

On a Product Formula for Unitary Groups

For any nonnegative self-adjoint operators A and B in a separableHilbert space, the Trotter-type formula is shown to converge strongly in the norm closureof dom (A^1/2) dom (B^1/2 for some subsequence and for almostevery t R. This result extends to the degenerate case, andto Kato-functions following the method of T. Kato (see ‘Trotter'sproduct formula for an arbitrary pair of self-adjoint contractionsemigroup’, Topics in functional analysis (ed. M. Kac,Academic Press, New York, 1978) 185–195). Moreover, therestrictions on the convergence can be removed by consideringfunctions other than the exponential. 2000 Mathematics SubjectClassification 47D03 (primary), 47B25 (secondary).     

Co-local subgroups of abelian groups II

In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] the notion of a co-local subgroup of an abelian group was introduced. A subgroup K of A is called co-local if the natural map is an isomorphism. At the center of attention in [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] were co-local subgroups of torsion-free abelian groups. In the present paper we shift our attention to co-local subgroups K of mixed, non-splitting abelian groups A with torsion subgroup t(A). We will show that any co-local subgroup K is a pure, cotorsion-free subgroup and if D/t(A) is the divisible part of A/t(A)=D/t(A)⊕H/t(A), then K∩D=0, and one may assume that K⊆H. We will construct examples to show that K need not be a co-local subgroup of H. Moreover, we will investigate connections between co-local subgroups of A and A/t(A).     

Rational torus-equivariant stable homotopy I: Calculating groups of stable maps

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show that it is of finite injective dimension. It can be used as a model for rational G-spectra in the sense that there is a homology theory

     

Every Set has a Least Jump Enumeration

Given a computably enumerable set W, there is a Turing degreewhich is the least jump of any set in which W is computablyenumerable, namely 0'. Remarkably, this is not a phenomenonof computably enumerable sets. It is shown that for every subsetA of N, there is a Turing degree, c'_µ(A), which is theleast degree of the jumps of all sets X for which A is . In addition this result providesan isomorphism invariant method for assigning Turing degreesto certain torsion-free abelian groups.     

Replacement of Factors by Subgroups in the Factorization of Abelian Groups

The above-titled paper of mine appeared in the Bulletin of theLondon Mathematical Society, 32 (2000) 297–304. Regrettably,there is a careless error in the proofs of Theorems 6 and 8.In line 6 of the proof of Theorem 6, it is claimed that a certainsubset must be a subgroup. For this to hold, the subset mustcontain the zero element. This need not be the case; the truededuction is that the subset is a coset, say M + h, of a subgroupM. Now M and M + h contain the same number of elements, andso the deduction that M has p elements is still correct. Similarly, in the proof of Theorem 8, the subgroup M_k must bereplaced by a coset M_k + h_k. This is the only change neededin this proof, since the sum M_k+h_k+(nBH) being direct impliesthat the sum M_k+(nBH) is also direct. Since the zero elementdoes belong to the sets (mAH) and (nBH), the statements aboutthese sets are correct. So the second paragraph of the Proofof Theorem 8 is correct, and is also a proof of Theorem 6. Now we present an example that, we hope, will clarify the situation,as well as showing that certain statements in the original ‘Proof’of Theorem 6 not only could be wrong but actually are wrong.The smallest numerical example occurs with p = 2, m = 3, n =5. Then G is a cyclic group of order 60, and may be representedas the integers modulo 60. Let A = {0, 1, 2, 3, 4} + {0, 15} and B = {0, 5, 10} + {0, 30}.It is easily verified that A + B = {0, 1,..., 59}. In the notationof Theorem 6, we see that H = {0, 15, 30, 45}, K = {0, 12, 24,36, 48}, L = {0, 20, 40}, and M = {0, 30}. Now we see that mA= {0, 3, 6, 9, 12} + {0, 45} M + K, and that nB = {0, 25, 50}+ {0, 30} M + L. We note, however, that A is a complete setof residues modulo 10; that is, that B can be replaced by M+ L.     

Continuous Bundles of C*-Algebras with Discontinuous Tensor Products

For each non-exact C*-algebra A and infinite compact Hausdorffspace X there exists a continuous bundle B of C*-algebras onX such that the minimal tensor product bundle AB is discontinuous.The bundle B can be chosen to be unital with constant simplefibre. When X is metrizable, B can also be chosen to be separable.As a corollary, a C*-algebra A is exact if and only if A Bis continuous for all unital continuous C*-bundles B on a giveninfinite compact Hausdorff base space. The key to proving theseresults is showing that for a non-exact C*-algebra A there existsa separable unital continuous C*-bundle B on [0,1] such thatA B is continuous on [0,1] and discontinuous at 1, a counter-intuitiveresult. For a non-exact C*-algebra A and separable C*-bundleB on [0,1], the set of points of discontinuity of A B in [0,1]can be of positive Lebesgue measure, and even of measure 1.2000 Mathematics Subject Classification 46L06 (primary), 46L35(secondary).     

Strength of convergence and multiplicities in the spectrum of a C*-dynamical system

We consider separable C*-dynamical systems (A, G,) for whichthe induced action of the group G on the primitive ideal spacePrim A of the C*-algebra A is free. We study how the representationtheory of the associated crossed product C*-algebra A G dependson the representation theory of A and the properties of theaction of G on Prim A and the spectrum Â. Our main toolsinvolve computations of upper and lower bounds on multiplicitynumbers associated to irreducible representations of A G. Weapply our techniques to give necessary and sufficient conditions,in terms of A and the action of G, for AG to be (i) a continuous-traceC*-algebra, (ii) a Fell C*-algebra and (iii) a bounded-traceC*-algebra. When G is amenable, we also give necessary and sufficientconditions for the crossed product C*-algebra AG to be (iv)a liminal C*-algebra and (v) a Type I C*-algebra. The resultsin (i), (iii)–(v) extend some earlier special cases inwhich A was assumed to have the corresponding property.