Symmetric Groups, Wreath Products, Morita Equivalences, and Broue's Abelian Defect Group Conjecture

It is shown that for any prime p, and any non-negative integerw less than p, there exist p-blocks of symmetric groups of defectw, which are Morita equivalent to the principal p-block of thegroup S_p S_w. Combined with work of J. Rickard, this provesthat Broué's abelian defect group conjecture holds forp-blocks of symmetric groups of defect at most 5.     

无限循环群被有限生成Abel群的中心扩张

设G是无限循环群被有限生成Abel群的中心扩张,T是G的中心ζG的挠子群.如果T的阶与ζG/(G'⊕T)的挠子群的阶互素,那么群G可分解为G=S×F×T,其中S= 这里d_i都是正整数,满足d_1|d_2|…|d_r,F是秩为s的自由Abel群,T是有限Abel群,T=Z_(e_1)⊕Z_(e_2)⊕…⊕Z_e_t,e_11,满足e_1|e_2|…|e_t,并且(d_1,e_t)=1.进一步,(d_1,d_2,…,d_T;s;e_1,e_2,…,e_t)是群G的同构不变量,即若群H也是无限循环群被有限生成Abel群的中心扩张,T_H是ζH的挠子群.如果T_H的阶与ζH/(H'⊕T_H)的挠子群的阶互索,那么G同构于H的充要条件是它们有相同的不变量.显然,这个结果涵盖了有限生成Abel群的结构定理.     

Random Walks on Wreath Products of Groups

We bound the rate of convergence to uniformity for a certain random walk on the complete monomial groups GS _n for any group G. Specifically, we determine that n log n+ n log (|G|–1|) steps are both necessary and sufficient for ² distance to become small. We also determine that n log n steps are both necessary and sufficient for total variation distance to become small. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group ₂S _n , the generalized symmetric group _m S _n , and S _m S _n . In the special case of the hyperoctahedral group, our random walk exhibits the cutoff phenomenon.     

Visibility of Shafarevich-Tate Groups of Abelian Varieties

We investigate Mazur's notion of visibility of elements of Shafarevich-Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich-Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton-Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich-Tate groups.     

Characteristics of Random Walks on Wreath Products of Groups

The paper analyzes the entropy and the normalized drift of the wreath product of finitely generated groups. Examples of groups of zero entropy are given. Bibliography: 4 titles.     

Direct Products with Abelian Automorphism Groups

The article considers when the direct product of two finite groups has an Abelian automorphism group.     

Abelian Subgroups of Finitely Generated Kleinian Groups are Separable

By a Kleinian group we mean a discrete subgroup of PSL(2, C).We prove that abelian subgroups of finitely generated Kleiniangroups are separable. In other words, if M = H³/ is a hyperbolic3-orbifold, with finitely generated, then abelian subgroupsof are separable in . 1991 Mathematics Subject Classification20E26, 51M10, 57M05.     

Fusion Procedure for Wreath Products of Finite Groups by the Symmetric Group

Let G be a finite group. A complete system of pairwise orthogonal idempotents is constructed for the wreath product of G by the symmetric group by means of a fusion procedure, that is by consecutive evaluations of a rational function with values in the group ring. This complete system of idempotents is indexed by standard Young multi-tableaux. Associated to the wreath product of G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is defined and appears in the rational function used in the fusion procedure.     

Symmetric Groups as Products of Abelian Subgroups

A proof is given that the full symmetric group over any infiniteset is the product of finitely many Abelian subgroups. In fact,289 subgroups suffice. Sharp bounds are also obtained on theminimal number k, such that the finite symmetric group S_n isthe product of k Abelian subgroups. Using this, S_n is provedto be the product of 72n^1/2(log n)^3/2 cyclic subgroups. 2000Mathematics Subject Classification 20B30, 20D40.     

The Automaton Realization of Iterated Wreath Products of Cyclic Groups

We study profinite groups which are infinitely iterated wreath products W _∞ = …?C _{n 2} ?C _{n 1} of finite cyclic groups via combinatorial language of transducers. Namely, we provide a naturally defined automaton realization of the group W _∞ by an automaton over a changing alphabet. Our construction gives a characterization of these profinite groups as automaton groups, i.e. as groups generated by a single automaton.     

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (\mathbbZ/l\mathbbZ)^m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.     

Probabilistic Generation of Wreath Products of Non-abelian Finite Simple Groups

Abstract

We show that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated. Information depending on the Classification of Finite Simple Groups is used throughout.     

Conjugacy Separability of Descending HNN-Extensions of Finitely Generated Abelian Groups

It is proved that an arbitrary descending HNN-extension of a finitely generated Abelian group is conjugacy separable.     

一类圈积的自同构群的矩阵描述

使用Bidwell和Curran在2006年引入的描述半直积的自同构的矩阵方法,结合作者等人在2010年证明的关于稳定自同构群的矩阵公式,得到了一类正则圈积的自同构群的矩阵描述,并求出了自同构群的阶.     

IA-Automorphisms of Metabelian Products of Two Abelian Groups

For metabelian products of two Abelian groups with torsion, IA-automorphisms that are not inner and hence are not induced by any automorphisms of the free products of the same Abelian groups are constructed.     

An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety [𝔑 _{c 1} , 𝔑 _{c 2} ], for all c ₂ ≤ c ₁ ≤ 2c ₂. As a consequence, we determine necessary and sufficient conditions for such groups to be [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable. We also show that if c ₁ ≠ 1 ≠ c ₂, then a finitely generated abelian group is [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable if and only if it is capable. Finally, we show that 𝔖₂-capability implies capability, but there is a capable finitely generated abelian group which is not 𝔖₂-capable.     

Computation of Irreducible Decompositions of Permutation Representations of Wreath Products of Finite Groups

Computational Mathematics and Mathematical Physics - An algorithm for the computation of the complete set of primitive orthogonal idempotents of the centralizer ring of the permutation...     

Semicomplete Permutational Wreath Products

A group is called semicomplete if every automorphism which induces the identity on the factor commutator group is inner. In this paper, we study the connection of the semicompleteness of the permutational wreath product W of two groups with the semicompleteness of these groups. We give necessary conditions under which the group W is semicomplete.2000 Mathematics Subject Classification: 20E22, 20E36