Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a u-product G₁ ○ G₂ of two u-groups G₁ and G₂, and prove that G₁ ○ G₂ is also a u-group and that every u-group, which contains G₁ and G₂ as subgroups and is generated by these, is a homomorphic image of G₁ ○ G₂. It is stated that if G is a u-group then the coordinate group of an affine space Gⁿ is equal to G ○ F_n, where F_n is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 601–621, September–October, 2005. Supported by RFBR grant No. 05-01-00292, by FP “Universities of Russia” grant No. 04.01.053, and by RF Ministry of Education grant No. E00-1.0-12.     

The property of being equationally Noetherian for some soluble groups

Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A₁ ⩾ A₂ ⩾ … A_n ⩾ … such that ⋂A_n = 1 and all factors A_n/A_n+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 46–59, January–February, 2007.     

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.     

Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G₂ and F₄. Supported by RFFR grant No. 96-01-01893, the program “Universities of Russia,” and by International Science Foundation and Government of Russia grant No. RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 663–684, November–December, 1996.     

Recognizability of finite simple groups <Emphasis Type="Italic">L</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) and <Emphasis Type="Italic">U</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) by spectrum

It is proved that finite simple groups L₄(2^m), m ⩾ 2, and U₄(2^m), m ⩾ 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups. As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order 8. Supported by RFBR (grant Nos. 05-01-00797 and 06-01-39001), by SB RAS (Complex Integration project No. 1.2), and by the Ministry of Education of China (Project for Retaining Foreign Expert). Supported by NSF of Chongqing (CSTC: 2005BB8096). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 83–93, January–February, 2008.     

Recognition of finite simple groupsL 3(2 m ) ANDU 3(2 m ) by their element orders

It is proved that a finite group isomorphic to a simple non-Abelian group L₃(2^m) or U₃(2^m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S₄(2^m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1. Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by the State Education Ministry of China (grant No. 98083). Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000.     

Periodic groups saturated by finite simple groups <Emphasis Type="Italic">U</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups in a set {U₃(2^m) | m = 1, 2, …} is isomorphic to U₃(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008.     

Intertwining operators and soliton equations

The fermionic approach to the Kadomtsev-Petviashvili hierarchy, suggested by the Kyoto school (Sato, Date, Jimbo, Kashiwara, and Miwa) in 1981–4, is generalized on the basis of the idea that, in a sense, the components of intertwining operators are a generalization of free fermions forgl _∞. Integrable hierarchies related to symmetries of Kac-Moody algebras are described in terms of intertwining operators. The bosonization of these operators for various choices of the Heisenberg subalgebra is explicity written out. These various realizations result in distinct hierarchies of soliton equations. For example, forsl _N -symmetries this gives the hierarchies obtained by the (n ₁,...,n _s )-reduction from thes-component KP hierarchy introduced by Kac and van de Leur. The research of both authors was supported in part by RFBR grant No. 96-02-18046, grant No. 96-15-96455 for the support of leading scientific schools, and RFBR-CNRS grant No. 98-01-22033. International Center for Nonlinear Science at the Landau Institute of Theoretical Physics. Institute of Theoretical and Experimental Physics. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 33, No. 4, pp. 1–24, October–December, 1999. Translated by M. I. Golenishcheva-Kutuzova     

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.     

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.     

Intermediate subgroups of chevalley groups over a field of fractions of a ring of principal ideals

Let K be a field of fractions of a principal ideal ring R and G_K be a Chevalley group (of normal type) over K. For each subring P ⊂ K, denote by G_P a subgroup of all elements of G_K with coefficients in P. Let M be intermediate between G_R and G_K, i.e., G_R ⊆ M ⊆ G_K. We prove that M=G_P for some intermediate subring P (R ⊆ P ⊆ K). Supported by RFFR grant No. 96-01-00409. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 347–358, May–June, 2000.     

Groups with an element permuting with finitely many of its conjugates

We study the problem as to which is the cardinality of connected components of the graph Γ_α, defined as follows. Let G be a group and a an element of G. The vertex set V(Γ_α) of the graph is the conjugacy class of elements,Cl _G(a), and two vertices x and y of the graph Γ_α are bridged by an edge iff x=y. If the intersectionC _G(a)∩Cl _G(a) is finite, Γ_α is locally finite. We prove that connected components of the locally finite graph Γ_α are finite in some classes of groups. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 543–551, September–October, 1996.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

Conjugate biprimitive finite groups saturated with finite simple subgroups

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 biprimitive finite groups saturated with groups of the sets {L₂(pⁿ)}, {Re(3²ⁿ⁺¹)}, and {Sz(2²ⁿ⁺¹)}. It is proved thai such groups are all isomorphic to {L₂(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 224–245, March–April, 1998.     

Characterizations of finite groups by sets of orders of their elements

For a finite group G, ω(G) denotes the set of orders of its elements. If ω is a subset of the set of natural numbers, h(ω) stands for the number of pairwise nonisomorphic finite groups G for which ω(G)=ɛ. We prove that h(ω(G))=1, if G is isomorphic to S₉, S₁₁, S₁₂, S₁₃, or A₁₂, and h(ω(G))=2 if G is isomorphic to S₂(6) or to O ₈ ⁺ (2). 01 Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 37–53, January–February, 1997.     

Primitive systems of elements in the varieties A m A n : Criterion and inducing

LetA _n, n≥0, be a variety of all Abelian groups whose exponental divides n. We establish a criterion of being primitive for varieties of the formA _m A _n, and study into the question of inducing primitive systems of elements in free groups of these. The results obtained give a solution to the problem by Bachmuth and Mochizuki concerning the tame range of varietiesA _m A _n for the case where m is freed of squares, and lend support to the conjecture by Bryant and Gupta as to inducing primitive, systems for varieties likeA _pnA. This author’s part is supported by RFFR grant No. 99-01-00567. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 513–530, September–October, 1999.     

The intersection of Sylow subgroups in finite groups

In Theorem 1, letting p be a prime, we prove: (1) If G=S_n is a symmetric group of degree n, then G contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 2), (2, 4), (2, 8)}, and (2) If H=A_n is an alternating group of degree n, then H contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 4)}. In Theorem 2, we argue that if G is a finite simple non-Abelian group and p is a prime, then G contains a pair of Sylow p-subgroups with trivial intersection. Also we present the corollary which says that if P is a Sylow subgroup of a finite simple non-Abelian group G, then ‖G‖>‖P‖². Supported by RFFR grants Nos. 93-01-01529, 93-01-01501, and 96-01-01893, and by International Science Foundation and Government of Russia grant RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 424–432, July–August, 1996.     

Strong Mal’tsev conditions in varieties of Abelian groups

Let Ab_d be a variety of Abelian groups of a finite exponent d≥1 and SC (Ab_d) be the set of all strong Mal’tsev conditions satisfied in Ab_d. We define the concept of a η-basis in SC(Ab_d) in terms of a basis w.r.t. a class η of varieties with commutative operations. The algorithm for constructing η-bases of any finite length in SC(Ab_d) is presented. For the variety Ab of all Abelian groups, we specify absolute bases of length 2 in SC(Ab) which are simultaneously η-bases. Bases of length 2 with similar properties are constructed also in SC(Ab_d), for any natural number d≥2. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 723–742, November–December, 1999.     

Dlab groups

We argue that for any subgroup H of rank 1 in a multiplicative group of positive reals, among Dlab groups of the closed intervalI=[0],[1] on an extended set of reals, there exist groups D_H*(I) and D_H* which lack normal relatively convex subgroups, are not simple groups, and have just two distinct linear orders. The cardinality of a set of linear orders on Dlab groups is computed. It is established that every rigid l-group is Abelian if it belongs to a varietyD of l-groups groups generated by the linearly ordered groups D_H*(I) and D_H*. We prove that the quasivariety q(D_H*(I), D_H*) of groups generated by D_H*(I) and D_H* is distinct from a quasivarietyO of all orderable groups. Similar results are stated for a variety of l-groups and the quasivariety of groups that are not embeddable in D_H*(I) and D_H*. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 531–548, September–October, 1999.