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.     

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.     

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.     

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.     

Quasirecognizability by the Set of Element Orders for Groups 3 D 4(q), for q Even

It is proved that if G is a finite group with an element order set as in the simple group ³D₄(q), where q is even, then the commutant of G/F(G) is isomorphic to ³D₄(q) and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 3–19, January–February, 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.     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

Periodic groups saturated with <Emphasis Type="Italic">L</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let be a set of finite groups. A group G is saturated with groups from if every finite subgroup of G is contained in a subgroup isomorphic to some member of . It is proved that a periodic group G saturated with groups from the set {L₃(2^m)|m = 1, 2, …} is isomorphic to L₃(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007.     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

Recognition of finite groups by a set of orders of their elements

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGL_n(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S₄(7) is recognizable, whereas A₁₀, U₃(5), U₃(7), U₄(2), and U₅(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A₅, A₇, A₈, A₉, A₁₁, A₁₂, L₂(q), q=7, 8, 11, 49, L₃(4), S₄(7), U₄(3), U₆(2), M₁₁, M₁₂, M₂₂, HS, or M^cL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A₆, A₁₀, U₃(3), U₄(2), U₅(2), U₃(5), or J₂, and G is nonrecognizable; (iii) G is isomorphic to S₆(2) or O ₈ ⁺ (2), and h(ω(G))=2. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.     

Group-theoretical generalization of necklace polynomials

Let G be a group, U a subgroup of G of finite index, X a finite alphabet and q an indeterminate. In this paper, we study symmetric polynomials M _G (X,U) and M_G^q(X,U)M_{G}^{q}(X,U) which were introduced as a group-theoretical generalization of necklace polynomials. Main results are to generalize identities satisfied by necklace polynomials due to Metropolis and Rota in a bijective way, and to express M_G^q(X,U)M_{G}^{q}(X,U) in terms of M _G (X,V)’s, where [V] ranges over a set of conjugacy classes of subgroups to which U is subconjugate. As a byproduct, we provide the explicit form of the GL _m (ℂ)-module whose character is M_\mathbbZ^q(X,n\mathbbZ)M_{\mathbb{Z}}^{q}(X,n\mathbb{Z}), where m is the cardinality of X.     

The quasivariety generated by a torsion-free Abelian-by-finite group

Let L_q(qG) be the quasivariety lattice contained in a quasivariety generated by a group G. It is proved that if G is a finitely generated torsion-free group in (i.e., G is an extension of an Abelian group by a group of exponent 2ⁿ), which is a split extension of an Abelian group by a cyclic group, then the lattice L_q(qG) is a finite chain. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 407–427, July–August, 2007.     

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.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

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.     

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.     

Soluble minimal non-(finite-by-Baer)-groups

Let \mathfrakX{\mathfrak{X}} be a class of groups. A group G is called a minimal non- \mathfrakX{\mathfrak{X}}-group if it is not an \mathfrakX{\mathfrak{X}}-group but all of whose proper subgroups are \mathfrakX{\mathfrak{X}}-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G ^′′ is a minimal non-nilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ _n (G′) is a minimal non-(finite-by-abelian)-group.     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.