Hyperzyklische gruppen

Definition: (a)G is called hypercyclic «iff each epimorphic imageH≠1 ofG possesses a cyclic normal subgroupA≠1». (b)G is called hypercentral «iff each epimorphic imageH≠1 ofG hasZ(H)≠1». (c) the set of prime numbers which divide the orders of the torsion elements (≠1) ofG is called «the characteristic ofG». Baer has shown that each hypercyclic groupG is a subdirect product of hypercyclic groups of finite characteristic. In this note we will characterize hypercentral groups by abelian torsion groups of finite exponent.     

Spectral Synthesis in Fourier Algebras of Ultrapherical Hypergroups

Let H be an ultraspherical hypergroup associated to a locally compact group G and a spherical projector π (in Muruganandam, Math. Nachr. 281:1590–1603, 2008). We investigate spectral synthesis properties of the Fourier algebra A(H), partly building on results for A(G). Special emphasis is placed on double coset hypergroups. We also present several examples displaying the diverse behavior of hypergroups in contrast to groups.     

Conjugation in Brauer algebras and applications to character theory

The Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define Brauer characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.     

Quasisimple classical groups and their complex group algebras

Let H be a finite quasisimple classical group, i.e., H is perfect and S:= H/Z(H) is a finite simple classical group. We prove that, excluding the open cases when S has a very exceptional Schur multiplier such as PSL₃(4) or PSU₄(3), H is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.     

On permutation characters of wreath products

It is known that the character rings of symmetric groups S_n and the character rings of hyperoctahedral groups S₂?S_n are generated by (transitive) permutation characters. These results of Young are generalized to wreath products G?H (G a finite group, H a permutation group acting on a finite set). It is shown that the character ring of G?H is generated by permutation characters if this holds for G, H and certain subgroups of H. This result can be sharpened for wreath products G?S_n;if the character ring of G has a basis of transitive permutation characters, then the same holds for the character ring of G?S_n.     

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ?,?,? were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.     

Hopf powers and orders for some bismash products

The exponent of a finite group G can be viewed as a Hopf algebraic invariant of the group algebra H=kG: it is the least integer n for which the nth Hopf power endomorphism [n] of H is trivial. The exponent of a group scheme G as studied by Gabriel and Tate and Oort can be defined in the same way using the coordinate Hopf algebra H=O(G).The power map and the corresponding notion of exponent have been studied for a general finite-dimensional Hopf algebra beginning with work of Kashina. Several positive results, suggested by analogy to the group case, were proved by Kashina and by Etingof and Gelaki.Given these positive results, there was some hope that the Hopf order of an individual element of a Hopf algebra might also be a well-behaved notion, with some properties analogous to well-known facts on the orders of elements of a finite group.In fact we prove that such analogous properties do hold for Hopf algebras satisfying the usual rule for iterated powers; for example, such a Hopf algebra H has an element of order n if and only if n divides the exponent of H. However, in general such properties are not true. We will give examples where the behavior of Hopf powers, Hopf orders, and related notions is rather strange, unexpected, and seemingly hard to predict. We will see this using computer algebra calculations in Drinfeld doubles of finite groups, and more generally in bismash products constructed from factorizable groups.     

Topological Hypergroups in the Sense of Marty

In this paper, we introduce the concept of topological hypergroups as a generalization of topological groups. A topological hypergroup is a nonempty set endowed with two structures, that of a topological space and that of a hypergroup. Let (H, ○) be a hypergroup and (H, τ) be a topological space such that the mappings (x, y) → x ○ y and (x, y) → x/y from H × H to 𝒫*(H) are continuous. The main tool to obtain basic properties of hypergroups is the fundamental relation β*. So, by considering the quotient topology induced by the fundamental relation on a hypergroup (H, ○) we show that if every open subset of H is a complete part, then the fundamental group of H is a topological group.

It is important to mention that in this paper the topological hypergroups are different from topological hypergroups which was initiated by Dunkl and Jewett.     

Lie rings admitting an automorphism of order 4 with few fixed points. II

We consider a Lie ring (algebra) L that admits an automorphism φ of order 4 with a finite number m of fixed points (with a fixed-point subalgebra of finite dimension m). It is proved that L contains a subring S of m-bounded index in the additive group L (a subalgebra S of m-bounded codimension), which possesses a nilpotent ideal I of class bounded by some constant, such that the factor-ring S/I is nilpotent of class ≤2. As a consequence, it is proved that, under the same conditions, L has a subring G of m-bounded index in the additive group of L (a subalgebra G of m-bounded codimension), in which an ideal generated by the Lie subring [G, ?²]=«ng?g+g^{? 2} | g∈G»ng (the subalgebra [G, ?²]=«ng?g+g^{? 2} | g∈G»ng is an ideal in G which) is nilpotent of class bounded by some constant (and its factor-algebra G/[G, ?²] is nilpotent of class ≤2 with a derived algebra (square) of m-bounded dimension). In proofs, we use the results of [1] and develop further the version of the method of generalized centralizers employed therein.     

The Brauer-Clifford Group for (S,H)-Azumaya Algebras over a Commutative Ring

The Brauer-Clifford group BrClif(Z,G) corresponding to a finite group G and a finite-dimensional semisimple G-algebra Z was recently introduced by Alexandre Turull in the course of his work on character correspondence conjectures in group representation theory. This Brauer-Clifford group is a group of equivalence classes of Azumaya algebras over Z whose G-algebra structure agrees on restriction to the fixed (and usually nontrivial) G-algebra structure of Z. In this paper we extend the notion of the Brauer-Clifford group to the case of (S,H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is a commutative H-module algebra. These Brauer-Clifford groups turn out to be an example of the Brauer group of the symmetric monoidal category of S # H-modules, a perspective which allows one to construct a dual Brauer-Clifford group for the category of S-modules with compatible right H-comodule structure.     

Recognition of characteristically simple group <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript> × <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript> by character degree graph and order

The character degree graph of a finite group G is the graph whose vertices are the prime divisors of the irreducible character degrees of G and two vertices p and q are joined by an edge if pq divides some irreducible character degree of G. It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple groups which are characterizable by this method.We prove that the characteristically simple group A₅ × A₅ is uniquely determined by its order and its character degree graph. We note that this is the first example of a non simple group which is determined by order and character degree graph. As a consequence of our result we conclude that A₅ × A₅ is uniquely determined by its complex group algebra.     

Fusions of Character Tables and Schur Rings of Abelian Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. The theory is developed in terms of the S-rings of Schur and Wielandt. We discuss certain classes of p-groups which fuse from abelian groups and give examples of such groups which do not. We also show that a large class of simple groups do not fuse from abelian groups. The methods to show fusion include the use of extensions which are Camina pairs, but other techniques on S-rings are also developed.     

On a class of wreath products of hypergroups and association schemes

We characterize finite hypergroups S (in the sense of Frédéric Marty in Huitième Congres des Mathématiciens, pp. 45–59, 1934) satisfying |pq|=1 for any two elements p and q in S with p≠q ^? in terms of wreath products. The result applies to association schemes of finite valency and provides a corresponding characterization in scheme theory. For association schemes S of finite valency satisfying the above condition, we provide a second characterization, a characterization in terms of the subconstituent algebra of S.     

Table algebras with central fusions

The character theory of table algebras is not as good as the character theory of finite groups. We introduce the notion of a table algebra with a central-fusion, in which the character theory has better properties. We study conditions under which a table algebra (A,B) has a central-fusion, and its central-fusion is exactly isomorphic to the wreath product of the central-fusion of a quotient table algebra of (A,B) and another table algebra. As a consequence, we obtain a complete characterization of table algebras with exactly one irreducible character whose degree and multiplicity are not equal. Applications to association schemes are also discussed.     

Hecke algebras associated with induced representationsLes algèbres de Hecke associées aux représentations induites

We define the Hecke von Neumann algebra $\text{L}\text{(G,H,σ)}$ associated with a group G, a subgroup H and a unitary representation σ of H. We show that when σ is finite dimensional, $\text{L}\text{(G,H,σ)}$ can be seen as a corner algebra of the tensor product of the group von Neumann algebra of a locally compact group and a matrix algebra. To cite this article: R. Curtis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 31–35     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative.     

Sous-groupes canoniques partiels

The reduction of Siegel varieties modulo a prime number p is stratified by the multiplicative rank of the p-divisible group of the universal abelian variety. For r ≥ 0, the maximal multiplicative subgroup of the restriction of the p-torsion group of the universal abelian variety to the r-th stratum lifts canonically to the tube of this stratum and defines a «partial canonical subgroup of rank r». We show that there exists a strict neighbourhood of the tube on which this subgroup extends in a finite flat way. On the ordinary stratum and its neighbourhood, we recover the usuel canonical subgroup studied by Abbes and Mokrane, and Andreatta and Gasbarri.     

Fusions of Character Tables II: p-Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. In a previous article, we gave examples of Camina pairs that fuse from abelian groups. In this article, we give more general examples of Camina triples that fuse from abelian groups. We use this result to give an example of a group which fuses from an abelian group, but which has a subgroup that does not. We also give an example of a powerful 2-group which does not fuse from an abelian group and of a regular 3-group which does not fuse from an abelian group.     

Block characters of the symmetric groups

A block character of a finite symmetric group is a positive definite function which depends only on the number of cycles in a permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of ${\mathfrak{S}}_{n}$ . The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group ${\mathfrak{S}}_{\infty}$ , along with their connection to the Thoma characters of the infinite linear group GL _∞(q) over a Galois field.