On pointed Hopf algebras associated with the symmetric groups

Let G be the symmetric group . It is an important open problem whether the dimension of the Nichols algebra is finite when is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs might give rise to finite-dimensional Nichols algebras. This work was partially supported by CONICET, ANPCyT and Secyt (UNC).     

Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories

Given an action of a finite group G on a fusion category we give a criterion for the category of G-equivariant objects in to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non-group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in [7]. Namely, we show that certain /2-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami [26] are equivalent to representation categories of non-group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.      

The absolute Galois groups of finite extensions of $${\mathbb{R}}(t)$$

Let R be a real closed field and L be a finite extension of R(t). We prove that Gal(L) ≅ Gal(R(t)) if L is formally real and Gal(L) is the free profinite group of rank card (R) if L is not formally real. Received: 3 April 2007     

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H _i } of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H _i have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union generates G can be replaced by the condition that it is equal to G). The work presented here is partially supported by NSF/DMS-0805932.     

Normalizers of p-subgroups in finite groups

In this paper we give a computational strategy for constructing the normalizer of a p-subgroup in a finite group. Received: 31 January 2008     

Trifactorized locally finite groups with min-p for every prime p

A group G is trifactorized if G = AB = AC = BC with three subgroups A, B and C of G. Some structural theorems about trifactorized locally finite groups with minimum condition on p-subgroups for every prime p are proved. For instance, it is shown that G is locally supersoluble (locally nilpotent) if A and B are locally nilpotent and C is locally supersoluble (locally nilpotent). The second and the third author wish to thank the Institute of Mathematics of the University of Mainz for their excellent hospitality during the preparation of this paper. The second author is grateful to the University of Stellenbosch, South Africa, and the third author to the Deutsche Forschungsgemeinschaft (DFG) for financial assistance.     

Automorphisms fixing elements of prime order in finite groups

Let σ be an automorphism of a finite group G and suppose that σ fixes every element of G that has prime order or order 4. The main result of this paper shows that the structure of the subgroup H=[G, σ] is severely limited in terms of the order n of σ. In particular, H has exponent dividing n and it is nilpotent of class bounded in terms of n.     

The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals

The result is: The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals is similar to the distribution of square-free numbers in short intervals. Moreover, a new estimate of the error term in the corresponding asymptotic formula is given, which improves former estimates.      

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included. This research of the second and third authors was partially supported by Serbian Ministry of Science and Environment, Grant No. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, grant ”Lattice methods and applications”.     

Extensions of covariantly finite subcategories

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg. This project is partially supported by China Postdoctoral Science Foundation (No.s 20070420125 and 200801230). The author also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong.     

Classically Normal Pure States

A pure state f of a von Neumann algebra is called classically normal if f is normal on any von Neumann subalgebra of on which f is multiplicative. Assuming the continuum hypothesis, a separably represented von Neumann algebra M has classically normal, singular pure states iff there is a central projection p ∈M such that pMp is a factor of type I_∞, II, or III.     

Note on limits of finite graphs

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.     

Product set estimates for non-commutative groups

We develop the Plünnecke-Ruzsa and Balog-Szemerédi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their centre. T. Tao is supported by a grant from the Packard Foundation.     

Convergence analysis of the high-order mimetic finite difference method

We prove second-order convergence of the conservative variable and its flux in the high-order MFD method. The convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients. For the case of non-constant coefficients, we also develop a new family of high-order MFD methods. Theoretical result are confirmed through numerical experiments.     

Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs

We characterize intrinsic regular submanifolds in the Heisenberg group as intrinsic differentiable graphs. G. Arena is supported by MIUR (Italy), by INDAM and by University of Trento. R. Serapioni is supported by MIUR (Italy), by GALA project of the Sixth Framework Programme of European Community and by University of Trento.     

A new family of maximal curves over a finite field

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type and defined over the finite field all have the maximum number of -rational points allowed by the Weil “explicit formulas”, and that these curves are -maximal curves over infinitely many algebraic extensions of . Serre showed that an -rational curve which is -covered by an -maximal curve is also -maximal. This has posed the problem of the existence of -maximal curves other than the Deligne–Lusztig curves and their -subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n ³ with n = p ^r  > 2, p ≥ 2 prime, we give a simple, explicit construction of an -maximal curve that is not -covered by any -maximal Deligne–Lusztig curve. Furthermore, the -automorphism group Aut has size n ³(n ³ + 1)(n ² − 1)(n ² − n + 1). Interestingly, has a very large -automorphism group with respect to its genus . Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni, PRIN 2006–2007.     

On $${\mathcal{F}}$$ -supplemented subgroups of finite groups

Let G be a finite group and a formation of finite groups. We say that a subgroup H of G is -supplemented in G if there exists a subgroup T of G such that G = TH and is contained in the -hypercenter of G/H _G . In this paper, we use -supplemented subgroups to study the structure of finite groups. A series of previously known results are unified and generalized. Research of the author is supported by a NNSF grant of China (Grant #10771180).     

Irreducible <Emphasis Type="Italic">p</Emphasis>-Brauer characters of <Emphasis Type="Italic">p</Emphasis>-power degree for the symmetric and alternating groups

Starting from the question when all irreducible p-Brauer characters for a symmetric or an alternating group are of p-power degree, we classify the p-modular irreducible representations of p-power dimension in some families of representations for these groups. In particular, this then allows to confirm a conjecture by W. Willems for the alternating groups. Received: 14 June 2006     

An isolic generalization of Cauchy's theorem for finite groups

In his note [5] Hausner states a simple combinatorial principle, namely:

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.