Sur la classification des automorphismes périodiques des C1-algèbres UHF

We obtain a complete classification up to conjugacy and up to outer conjugacy of finite tensor product type automorphisms of UHF C¹-algebras which are periodic of period N (N prime).     

Conjugacy Length in Group Extensions

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behavior of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ?^d ? ?^k have at most exponential (if k > 1) or linear (if k = 1) conjugacy length functions.     

On elements of order <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> in the plane Cremona group over a field of characteristic <Emphasis Type="Italic">p</Emphasis>

We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of order of power of p larger than 2. We also describe conjugacy classes of elements of order p ². To the memory of Vasily Iskovskikh     

The conjugacy vector of ap-group of maximal class

We find the conjugacy vector, i.e., we determine the number of conjugacy classes which compose the sets of the elements with centralizers of equal order, for several general families ofp-groups of maximal class which include those of order up top ⁹. As a consequence, we obtain the number of conjugacy classes,r(G), for the groups in these families. Also, we provide upper and lower bounds forr(G) and characterize when they are attained. Examples are given showing that the bounds are actually attained. This work has been supported by DGICYT grant PB91-0446 and by the University of the Basque Country.     

On Reflection Subgroups of Finite Coxeter Groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.     

Classification of topologically conjugate affine mappings

We consider affine mappings from ℝ ⁿ into ℝ ⁿ , n ≥ 1. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from R into R and also for affine mappings from ℝ ⁿ into ℝ ⁿ , n > 1, having at least one fixed point and the nonperiodic linear part. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 134–139, January, 2009.     

On the Validity of Thompson's Conjecture for Finite Simple Groups

In this article, we prove a conjecture of J. G. Thompson for the finite simple group ² D _n (q). More precisely, we show that every finite group G with the property Z(G) = 1 and N(G) = N(² D _n (q)) is necessarily isomorphic to ² D _n (q). Note that N(G) is the set of lengths of conjugacy classes of G.     

Ahlfors-régularité des quasi-minima de Mumford–Shah

Let be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the -norm of U term is removed, where K is a closed subset of Ω and U is a function on ΩK with gradient in  . The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the -measure of is bounded above by cr^N−1 and bounded below by c⁻¹r^N−1.     

The Dichotomy Problem for Orbital Measures of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)

For many orbital measures μ, on SU(n), we show that either μ^k ∈ L² or μ^k is singular to L¹. The least k for which μ^k ∈ L² is determined and is shown to be the minimum k for which the k-fold product of the conjugacy class supporting the measure has positive measure. It would be interesting to know if all orbital measures satisfy this dichotomy.     

On moduli of stability of two-dimensional vector fields

In this paper the author proves that, on a compact connected and orientable two-dimensionalC ^∞ manifoldM, the gradient of aC ³-Morse Function has finite modulus of stability under conjugacy and modulus zero under topological equivalence. It is also proved that generically the modulus of stability under conjugacy of a graph of aC ² vector field on the plane is at least twice the number of its saddles. Some new conjugacy invariants arise in the proofs of these results. During the preparation of this paper the author was a visiting Professor at IMPA; and was partially suported by Financiadora de Estudo e Projetos (FINEP).     

Conjugacy Classes Outside a Normal Subgroup

Abstract

In this paper, we investigate the structure of finite groups G admitting a normal subgroup N such that N contains almost all conjugacy classes of G.     

On the number of conjugacy classes of a finite solvable group II

We prove that a finite solvable group G has at least (49p+1)/60 conjugacy classes whenever p is a prime such that p² divides the order of G. We also construct an infinite family of finite solvable groups, where this bound is attained.     

Examples of C1-smoothly conjugate diffeomorphisms of the circle with break that are not C1+γ-smoothly conjugate

We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not C ^1+γ -smoothly conjugate for any γ > 0. In this way, we show that the previous result concerning the C ¹-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.     

Conjugacy Classes,Class Sums and Character Tables for Hopf Algebras

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d ^?2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS ₃), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.     

Maximal Complements in Finite Groups

Let G be a finite group with a non-Abelian minimal normal subgroup N which is a direct product of copies of the simple group X. A parametrization is given for the conjugacy classes of maximal subgroups of G which complement N in terms of certain homomorphisms taking values in Aut X.     

On the number of conjugacy classes in a finite group II

In this work we obtain new properties connected with the number of conjugacy classes of elements of a finite group, through the analysis of the numberr _G(gN) of conjugacy classes of elements ofG that intersect the cosetgN, whereN is a normal subgroup ofG andg any element ofG. The results obtained about this number are not only used in the general problem of classifying finite groups according to the number of conjugacy classes, but they also allow us to improve and generalize known results relating to conjugacy classes due to P. Hall, M. Cartwright, A. Mann, G. Sherman, A. Vera-López and L. Ortíz de Elguea. Examples are given which illustrate our improvements. This work has been supported by the University of the Basque Country.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

Krull–Schmidt reduction of principal bundles in arbitrary characteristic

Let M be an irreducible projective variety defined over an algebraically closed field k, and let E_G be a principal G-bundle over M, where G is a connected reductive linear algebraic group defined over k. We show that for E_G there is a naturally associated conjugacy class of Levi subgroups of G. Given a Levi subgroup H in this conjugacy class, the principal G-bundle E_G admits a reduction of structure group to H. Furthermore, this reduction is unique up to an automorphism of E_G.     

Effacement des dérivations et spectres premiers des algèbres quantiques

Given any (commutative) field k and any iterated Ore extension R=k[X₁][X₂;σ₂,δ₂][X_N;σ_N,δ_N] satisfying some suitable assumptions, we construct the so-called “Derivative-Elimination Algorithm.” It consists of a sequence of changes of variables inside the division ring F=Fract(R), starting with the indeterminates (X₁,…,X_N) and terminating with new variables (T₁,…,T_N). These new variables generate some quantum-affine space such that . This algorithm induces a natural embedding which satisfies the following property:

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. We study both the derivative-elimination algorithm and natural embedding and use them to produce, for the general case, a (common) proof of the “quantum Gelfand–Kirillov” property for the prime homomorphic images of the following quantum algebras: , (wW), R_q[G] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra and Weyl group W), quantum matrices algebras, quantum Weyl algebras and quantum Euclidean (respectively symplectic) spaces. Another application will be given in [G. Cauchon, J. Algebra, to appear]: In the general case, the prime spectrum of any quantum matrices algebra satisfies the normal separation property.     

On non-solvable Camina pairs

In this paper we study non-solvable and non-Frobenius Camina pairs (G,N). It is known [D. Chillag, A. Mann, C. Scoppola, Generalized Frobenius groups II, Israel J. Math. 62 (1988) 269–282] that in this case N is a p-group. Our first result (Theorem 1.3) shows that the solvable residual of G/O_p(G) is isomorphic either to SL(2,p^e),p is a prime or to SL(2,5), SL(2,13) with p=3, or to SL(2,5) with p7.Our second result provides an example of a non-solvable and non-Frobenius Camina pair (G,N) with |O_p(G)|=5⁵ and G/O_p(G)SL(2,5). Note that G has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studies by S.M. Gagola [S.M. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math. 109 (1983) 363–385]. To our knowledge this group is the first example of a Gagola group which is non-solvable and non-Frobenius.