On the Asymptotic Geometry of Nonpositively Curved Manifolds

In this paper we derive new asymptotic properties of all Hadamard manifolds admitting compact quotients. We study the growth function of the volume of geodesic spheres, generalizing the work of Margulis in the case of negative curvature. We show that the growth is of purely exponential type if and only if the Hadamard manifold is of rank 1. In general, there is a polynomial deviation from purely exponential behavior, depending in an unexpected way on the rank of the manifold. Furthermore, we obtain new results on the growth rate of closed geodesics on compact rank 1 spaces. Submitted: August 1996, final version: April 1997     

Conjugacy Classes and Characters for Extensions of Finite Groups

Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for tale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

Conjugacy Classes in Sylow p-Subgroups of Finite Chevalley Groups in Bad Characteristic

Let U = U(q) be a Sylow p-subgroup of a finite Chevalley group G = G(q). Röhrle and Goodwin in 2009 determined a parameterization of the conjugacy classes of U, for G of small rank when q is a power of a good prime for G. As a consequence they verified that the number k(U) of conjugacy classes of U is given by a polynomial in q with integer coefficients. In the present paper, we consider the case when p is a bad prime for G. Our motivation is to observe how the situation differs between good and bad characteristics. We obtain a parameterization of the conjugacy classes of U, when G has rank less than or equal to 4, and G is not of type F ₄. In these cases we deduce that k(U) is given by a polynomial in q with integer coefficients; this polynomial is different from the polynomial for good primes.     

On Lengths of Conjugacy Classes and Character Degrees in Finite Groups

We extend the two results of Dolfi and Isaacs to π-separable groups, where π = {p, q}, from solvable groups.     

Bounds for the Number of Conjugacy Classes of Non-Normal Subgroups in a Finite p-Group

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. We obtain two new lower bounds for ν(G) when G is a non-abelian finite p-group and p is odd. More precisely, if |G| =p ⁿ , exp Z(G) = p ^e , and exp G/G′ =p ^f , let us define λ(G) = n ? e and κ(G) = n ? f. Then we prove that ν(G) ≥ p(λ(G) ?3) +2 and ν(G) ≥ p(κ(G) ?3) +2. The first bound improves the bound ν(G) ≥ λ(G) ?1 given by [10 La Haye , R. , Rhemtulla , A. ( 1999 ). Groups with a bounded number of conjugacy classes of non-normal subgroups . J. Algebra 214 : 41 – 63 .[Crossref], [Web of Science ®] , [Google Scholar]], and almost in every case, the second one improves the bound ν(G) ≥ p(k ? 1) +1 obtained by [6 Fernández-Alcober , G. A. , Legarreta , L. ( 2008 ). Conjugacy classes of non-normal subgroups in finite nilpotent groups . J. Group Theory 11 ( 3 ): 381 – 397 .[Crossref] , [Google Scholar]], where k is defined by the condition that |G′| =p ^k .     

Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds

Let be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space H ⁿ⁺¹. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = \H ⁿ⁺¹, under the assumption that H ¹(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.     

On Finite Number of Conjugacy Classes in Groups

The work is inspired by an article of Herzog, Longobardi, and Maj, who considered groups with a finite number of infinite conjugacy classes. Their main results were obtained under assumption that the FC-center is of finite index in the group. We consider here infinite groups with a finite number of conjugacy classes of any size (FNCC-groups). Hence the FC-center in our case will be finite, but of infinite index in the group. Among results on these groups we give a criterion for a wreath product of FNCC-groups to be an FNCC-group.     

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.     

Conjugacy Classes and Invariant Subrings of R-Automorphisms of R[x]

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ?₂ and m ² = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.     

一类和有限群的共轭类关联的新图

给出了满足下列性质P_4的有限群的分类,对任意4个不同的非中心的共轭类,它们的代表元的阶的公因子为1.     

Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.     

关于群的阶与共轭类数

讨论群的共轭数与群阶的关系,获得两个新的数量不等式,同时改进了一些相关的已知结果。     

共轭类的长和有限群的结构

设Ｇ是有限群，并设Ｃｏｎ（Ｇ）是由Ｇ的一切共轭类组成的集合。本文目的是考察Ｃｏｎ（Ｇ）的算术结构对Ｇ的群结构的影响。我们着重于Ｃｏｎ（Ｇ）的ｐ－部分结构，并得到关于ｐ－幂零群的两个定理。这里，ｐ表示一个有限群的最小素因子。用这两个定理，我们还得到若干结果，其中两个改进了Ｄ．Ｃｈｉｌｌａｇ和Ｍ．Ｈｅｒｚｏｇ关于共轭类长的两个结果。     

Conjugacy of Sylow 2-Subloops of the Chein Loops M 2n (G, 2)

In general, Sylow's Theorems do not hold for finite Moufang loops. It can be seen that if p is an odd prime then the Sylow p-subloops of the Chein loop M _2n (G, 2) are conjugate. Here we prove that it is also true that all the Sylow 2-subloops of M _2n (G, 2) are conjugate.     

幂零群中非正规循环子群的共轭类数

设G是有限幂零群,v~*(G)是其非正规循环子群的共轭类数,则下列结论之一成立：(1) v~*(G)≥c(G)-1；(2)G是Hamilton群；(3)G中存在正规子群K使K/Z(K)有一个同态像与二面体群D(2~n),n≥3或C_2×C_2同构．     

Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost -simple algebraic group with non-compact and a cocompact congruence subgroup. For any homogeneous manifold of finite volume, and a , we show that the Hecke orbit T _a (x ₀ H) is equidistributed on as , provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial. The second author partially supported by DMS 0333397. Received: May 2005 Revision: March 2006 Accepted: June 2006     

Lorentz Geometry of 2-Step Nilpotent Lie Groups

We study the geometry of 2-step nilpotent Lie groups endowed with left-invariant Lorentz metrics. After integrating explicitly the geodesic equations, we discuss the problem of the existence of translated geodesics in those groups. A good part of the paper focuses on the existence of closed timelike geodesics in compact Lorentz 2-step nilmanifolds. Other related results, corollaries, and examples are also presented.