Periodic part in some Shunkov groups

A group G is saturated with groups in a set X if every finite subgroup of G is embeddable in G into a subgroup L isomorphic to some group in X. We show that a Shunkov group has a periodic part if the saturating set for it coincides with one of the following: {L₂(q)}, {Sz(q)}, {Re(q)}, or {U₃(2ⁿ)}. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 96–125, January–February, 1999.     

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.     

Some subgroups of semilinearly ordered groups

Let G be a semilinearly ordered group with a positive cone P. Denote byn(G) the greatest convex directed normal subgroup of G, byo(G) the greatest convex right-ordered subgroup of G, and byr(G) a set of all elements x of G such that x and x⁻¹ are comparable with any element of P^± (the collection of all group elements comparable with an identity element). Previously. it was proved thatr(G) is a convex right-ordered subgroup of G. andn(G) ⊆r(G) ⊆o(G). Here, we establish a new property ofr(G). and show that the inequalities in the given system of inclusions are, generally, strict. Supported by RFFR grant No. 99-01-00156. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 465–479, July–August, 2000.     

Distributivity conditions for lattices of dominions in quasivarieties of Abelian groups

Let ℳ be any quasivariety of Abelian groups, L_q(ℳ) be a subquasivariety lattice of ℳ, dom _G ^ℳ be the dominion of a subgroup H of a group G in ℳ, and G/dom _G ^ℳ (H) be a finitely generated group. It is known that the set L(G, H, ℳ) = {dom _G ^N (H)| N ∈ L_q(ℳ)} forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of dom _G ^ℳ (H). It is proved that the lattice L(G,H,ℳ) is semidistributive and necessary and sufficient conditions are specified for its being distributive. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 484–499, July–August, 2006.     

A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

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.     

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.     

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.     

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.     

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.     

On a width of verbal subgroups of certain free constructions

We study into widths of verbal subgroups of HNN-extensions, and of groups with one defining relation. It is proved that if a group G^* is an HNN-extension and the connected subgroups in G^* are distinct from a base of the extension, then every verbal subgroup V(G^*) has infinite width relative to a finite proper set V of words. A similar statement is proven to hold for groups presented by one defining relation and ≥3 generators. to Yurii I. Merzlyakov dedicated Supported by RFFR grant No. 93-01-01513. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 494–517, September–October, 1997.     

Cayley graphs of the group ℤ4 and limits for minimal vertex-primitive graphs of HA-type

We point out a countable set of pairwise nonisomorphic Cayley graphs of the group ℤ⁴ that are limit for finite minimal vertex-primitive graphs admitting a vertex-primitive automorphism group containing a regular Abelian normal subgroup. Supported by RFBR grant No. 06-01-00378. __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 203–214, March–April, 2008.     

The Characterization of Finite Simple Groups, L3（3^2m-1）（m≥2）, by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.     

Differentiation in the branges spaces and embedding theorems

The boundedness conditions for the differentiation operator in Hilbert spaces of entire functions (Branges spaces) and conditions under which the embedding K_и⊂L₂(μ) holds in spaces K_и associated with the Branges spacesH(E) are studied. Measure μ such that the above embedding is isometric are of special interest. It turns out that the condition E'/E∈H^∞(C⁺) is sufficient for the boundedness of the differentiation operator inH(E). Under certain restrictions on E, this condition is also necessary. However, this fact fails in the general case, which is demonstrated by the counterexamples constructed in this paper. The convex structure of the set of measures μ such that the embedding K_E ^* _/E⊂L²(μ) is isometric (the set of such measures was described by de Brages) is considered. Some classes of measures that are extreme points in the set of Branges measures are distinguished. Examples of measures that are not extreme points are also given. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 27–68.     

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.     

One-regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.     

On a problem from the Kourovka Notebook

In this article, it is proved that if a group G coincides with its commutator subgroup, is generated by a finite set of classes of conjugate elements, and contains a proper minimal normal subgroup A such that the factor group G/A coincides with the normal closure of one element, then G coincides with the normal closure of an element. From this a positive answer to question 5.52 from the Kourovka Notebook for the group with the condition of minimality on normal subgroups follows. We have found a necessary and sufficient condition for a group coinciding with its commutator subgroup and generated by a finite set of classes of conjugate elements not to coincide with the normal closure of any element. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 119–125, 2005.     

Subgroups of symplectic groups that contain a subsystem subgroup

Let Γ = GSp(2l, R) be the general symplectic group of rank l over a commutative ring R such that 2 ∈ R*; and let ν be a symmetric equivalence relation on the index set {1,…, l, −l,…, 1} all of whose classes contain at least 3 elements. In the present paper, we prove that if a subgroup H of Γ contains the group E_Γ(ν) of elementary block diagonal matrices of type ν, then H normalizes the subgroup generated by all elementary symplectic transvections T_ij(ξ) ∈ H. Combined with the previous results, this completely describes the overgroups of subsystem subgroups in this case. Similar results for subgroups of GL(n, R) were established by Z. I. Borewicz and the author in the early 1980s, while for GSp(2l, R) and GO(n, R) they have been announced by the author in the late 1980s, but a complete proof for the symplectic case has not been published before. Bibliography: 74 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 5–29.     

Normal relatively convex subgroups of solvable orderable groups

Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed than it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group G of length three with periodic G/G′ and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed. To the memory of N. Ya. Medvedev Supported by RFBR (project No. 03-01-00320). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 291–308, May–June, 2009.     

Relatively Compact Sets on Abstract Wiener Space

In this note, we obtain a sufficient and necessary condition for a set in an abstract Winner space （X, H, μ） to be relatively compact in L^2（X, μ）. Meanwhile, we give a sufficient condition for relative compactness in L^P（X, μ） for p〉1. We also provide an example of Da Prato-Malliavin Nualart to show the result.