Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)

Let Г_r,n—r denote the infimum of all number Г > 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c₁; c₂,…, c_n, there exist integersx ₁,x₂,…,x_n satisfying 0 < Q(x₁+c₁,x₂ + c₂,…,x_n + c_n) ≤(Г|Z > |)^1/n. All the values of Г_r,n—r are known except for г_1,4. Earlier it was shown that 8 ≤Г_1,4 ≤16. Here we improve the upper bound to get Г_1,4 < 12.     

The primitive permutation groups of degree 2 a ·3 b

In this paper, we give a complete classification of all finite simple groups with maximal subgroups of index n, where n = 2^a·3^b for a, b≧ 1. As a consequence, for such n, all primitive permutation groups of degree n are given. The motivation of this work comes also from a study of Cayley graphs of certain valency on a finite simple group. Received: 9 March 2005     

Characters of symmetric groups: sharp bounds and applications

We provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible. It follows from our general bound that if a permutation σ∈S _n has at most n ^o(1) cycles of length <m, then |χ(σ)|≤χ(1)^1/m+o(1) for all irreducible characters χ of S _n . This is a far reaching generalization of a result of Fomin and Lulov. We then use our various character bounds to solve a wide range of open problems regarding mixing times of random walks, covering by powers of conjugacy classes, as well as probabilistic and combinatorial properties of word maps. In particular we prove a conjecture of Rudvalis and of Vishne on covering numbers, and a conjecture of Lulov and Pak on mixing times of certain random walks on S _n . Our character-theoretic methods also yield best possible solutions to Waring type problems for alternating groups A _n , showing that if w is a non-trivial word, and n≫0, then every element of A _n is a product of two values of w.     

Universally equivalent solvable groups

Every group that is finitely presented in the varietyA ⁿ of solvable groups. and is universally equivalent to a free group F_r(A ⁿ) in this variety, is embedded in the Cartesian degree of F₂(A ⁿ). All subgroups on a set of two generators in that Cartesian degree which are universally equivalent to F₂(A ⁿ) are determined. Free solvable and nilpotent groups are proved universally equivalent. Supported by RFFR grant No. 99-01-00567, and through the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 227–240, March–April, 2000.     

Free subgroups of small cancellation groups

In this paper we give a simple proof of Collin’s theorem concerning free subgroups ofC(4),T(4) groups. Our proof actually shows that a slenderT(4) presentation 〈x ₁,x ₂, …,x _n ;r〉 has a free subgroup of rank 2 provided there is a subset {a, b, c} of {x ₁,x ₂, …,x _n } with the property that any non-empty freely reduced word ina, b, c equal to 1 inG has a subword of length 2 contained in an element ofr*.     

Fixed points of finite groups acting on generalised Thompson groups

We study centralisers of finite order automorphisms of the generalised Thompson groups F _n,∞ and conjugacy classes of finite subgroups in finite extensions of F _n,∞. In particular, we show that centralisers of finite automorphisms in F _n,∞ are either of type FP_∞ or not finitely generated.     

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

A minimal permutation representation of a group is its faithful permutation representation of least degree. We will find degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, for groups of types E₆, E₇, and E₈. This brings to a close the study of minimal permutation representations of finite simple Chevalley groups. Supported by RFFR grant No. 93-01-01501, through the program “Universities of Russia,” and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 518–530, September–October, 1997.     

Conjugate biprimitive finite groups saturated with finite simple subgroupsU 3(2 n )

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic conjugate biprimitive finite groups saturated with groups in the set {U₃(2ⁿ)}. It is proved that every such group is isomorphic to a simple group U₃(Q) over a locally finite field Q of characteristic 2. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 606–615, September–October, 1998.     

Inversion of matrices over a pseudocomplemented lattice

We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with and and let A = ‖a _ij ‖ _n×n , where a _ij ∈ P for i, j = 1,..., n. Let A* = ‖a _ij ^′ ‖ _n×n and for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤). Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL _n (P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) − , ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL _a (P, ≤) ≅ = S _n ^k . We give some further results concerning inversion of matrices over a pseudocomplemented lattice. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005.     

Central points for sets in ℝn (or: the chocolate ice-cream problem)

LetA be a subset of the unit ball in ℝⁿ, and let 0≤r≤1 be real. Find a pointx for which the intersection of ther-neighborhood ofx withA has a large measure. Tight bounds on this measure are found. This work was supported in part by grants from the Israeli Academy of Sciences, the Binational Science Foundation Israel-USA and the Niedersachsen-Israel Research Program.     

Maximal subgroups of odd index in finite groups with simple linear,unitary, or symplectic socle

We give a classification of maximal subgroups of odd index in finite groups whose socle is isomorphic to one of the groups PSL _n (q), PSU _n (q), or PSp _n (q) for n ≥ 13.     

Completely simple semigroups with nilpotent structure groups

In this paper we find simple characterizations of completely simple semigroups with H-classes nilpotent of class ≤c, and of completely simple semigroups whose core has H-classes nilpotent of class ≤c. The notion of w-marginal completely regular semigroups is introduced, generalizing the concept of central semigroups. A law characterizing [x ₁,x ₂,…,x _c+1]-marginal completely simple semigroups is obtained. Additionally, the least congruences corresponding to these classes are described. Our results extend the corresponding results obtained by Petrich and Reilly in the abelian case. The author was supported by the Ministry of Higher Education, Science and Technology of Slovenia.     

A Freiheitssatz for products of groups

Denote byB a class of solvable groups having a finite normal series with torsion-free Abelian factors, and by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaWefv3ySLgzgjxyRrxDYbqehuuDJXwAKbIrYf2A0vNC% aGGbaiqb-fa8czaaraaaaa!475E!\[\bar \mathfrak{B}\] a class of groups every finitely generated subgroup of which is approximated by {ie193-3}. We prove that if {ie193-4} is a free product with relations of groups A₁,…,A_n in the class {ie193-5}, where n>m and all relations are taken from the Cartesian subgroups, then there exist distinct indices i₁,…,i_n-m such that gp(A_i1,…,Ai_n-m)=A_i1 *…* Ai_n-m. A similar fact is established for solvable products with relations. Supported by RFFR grant No. 99-01-00567. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 354–367, May–June, 1999.     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

Isometric approximation

Suppose thatA ⊂R ⁿ is a bounded set of diameter 1 and that:f:A →l ₂ is a map satisfying the nearisometry condition |x−y|−ɛ≤|fx−fy|≤|x−y|+ɛ withɛ≤1. Then there is an isometryS:A →l ₂ such that |Sx−fx|≤c _n √ɛ for allx inA. IfA satisfies a thickness condition and iff:A →R ⁿ , then there is an isometryS:R ⁿ →R ⁿ with |Sx−fx|≤c _nɛ/q, whereq is a thickness parameter.     

Confined subgroups in periodic simple finitary linear groups

A subgroupX of the locally finite groupG is said to beconfined, if there exists a finite subgroupF≤G such thatX ^g∩F≠1 for allg∈G. Since there seems to be a certain correspondence between proper confined subgroups inG and non-trivial ideals in the complex group algebra ℂG, we determine the confined subgroups of periodic simple finitary linear groups in this paper. Dedicated to the memory of our friend and collaborator Richard E. Phillips     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

The ranks of central unit groups of integral group rings of alternating groups

Let G be a finite group and U(Z(Z G)) be the group of units of the center Z(Z G) of the integral group ring Z G (the central unit group of the ring Z G). The purpose of the present work is to study the ranks r _n of groups U(Z(ZAn)), i.e., of central unit groups of integral group rings of alternating groups A _n . We shall find all values n for r _n = 1 and propose an approach on how to describe the groups U(Z(ZAn)) in these cases, and we will present some results of calculations of r _n for n ≤ 600.     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

Analysis on discrete cocompact subgroups of the generic filiform Lie groups

Summary Let F_n, n≧ 1, denote the sequence of generic filiform (connected, simply connected) Lie groups. Here we study, for each F_n, the infinite dimensional simple quotients of the group C*-algebra of (the most obvious) one of its discrete cocompact subgroups D_n. For D_n, the most attractive concrete faithful representations are given in terms of Anzai flows, in analogy with the representations of the discrete Heisenberg group H₃ ⊆G₃ on L²(T) that result from the irrational rotation flows on T; the representations of D_n generate infinite-dimensional simple quotients A_n,θ of the group C*-algebra C*(D_n). For n>1, there are other infinite-dimensional simple quotients of C*(D_n) arising from non-faithful representations of D_n. Flows for these are determined, and they are also characterized and represented as matrix algebras over simple affine Furstenberg transformation group C*-algebras of the lower dimensional tori.