Torsion-free groups with factor-groups on their hypercenter which are periodic

Assume that G is a torsion-free group, Zk(G) is the k-th term of the upper central series of G, and ¯G_k=G/Z_k(G) is a nontrivial periodic group. Then every finite subgroup of ¯G_k is nilpotent of class not higher than k; the group k 2 contains an infinite subgroup with k generators if k2 and two generators if k=1. Moreover any nontrivial invariant subgroup of ¯G_k is infinite. All elements of ¯Gk are of odd order. This assertion is generalized.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 373–383, September, 1970.     

A GENERALIZATION OF COMPLETE GROUPS

Let G be a finite group and let G be the semi-direct product of a normal subgroup N and a subgroup K. In [1], conditions were found which are equivalent to the existence of a normal complement to N in G. We consider the structure of groups N for which the above condition always holds. Thus we use Bechtell's results to gain information on groups N such that if G is a semi-direct product of N and a subgroup K, then N is a direct factor of G, for all G. It is an old result that a group N is complete if and only if whenever N is a normal subgroup of G, then N is a direct factor of G, [4]. Hence it is not surprising that complete groups are part of our result. Moreover a group N is complete if and only if N is isomorphic to Aut(N) under the mapping σ(n) = σ _n , where σ _n is the inner automorphism induced by n. This remark leads us to consider groups N which contain a subgroup H such that H is isomorphic to Aut(N) under σ: H → Aut(N). All groups considered here are finite. The results found here do not parallel the results found in the author's dissertation for Lie algebras. There it is shown that only complete Lie algebras have the desired property. Thus, these results provide an example of when the theory of Lie algebras diverges from that of groups.     

Lattice of subgroups

Let D be a subgroup of the group G. The lattice of intermediate subgroups is studied. The subgroup F (D F G) is said to be D-complete, if D^F=u:u F>=F. Let F be the subset of all D-complete intermediate subgroups. The system {F, N_G(F)} is a fan for D in G (RZhMat, 1980, 5A208) if and only if D^x> is a D-complete subgroup for any x G. The set {F_ga} coincides with the collection of subgroups of the form D^A (1 A C G) if and only if for any x G the subgroup D, D^x is D-complete. The last condition holds, for example, for a pronormal subgroup D.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 13–19, 1980.     

Arrangement of subgroups

Suppose G is a group and D a subgroup. A system, of intermediate subgroups G and their normalizers is called a fan for D if for each intermediate sub group H (D HG) there exists a unique index such that. If there exists a fan for D, then D is called a fan subgroup of G. Examples of fans and fan subgroups are given. A standard fan is distinguished, for which all of the groups G are generated by sets of subgroups conjugate to D. The question of the uniqueness of a fan is discussed. It is proved that any pronormal subgroup is a fan subgroup, and some properties of its fan are noted.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 5–12, 1979.     

The Galois Correspondence in Field Algebra of G-spin Model

Suppose that G is a finite group and D（G） the double algebra of G. For a given subgroup H of G, there is a sub-Hopf algebra D（G; H） of D（G）. This paper gives the concrete construction of a D（G; H）-invariant subspace AH in field algebra of G-spin model and proves that if H is a normal subgroup of G, then AH is Galois closed.     

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.     

Lattices of dominions in quasivarieties of Abelian groups

Let M be any quasivariety of Abelian groups, (H) be the dominion of a subgroup H of a group G in M, and L_q(M) be the lattice of subquasivarieties of M. It is proved that (H ) coincides with a least normal subgroup of the group G containing H, the factor group with respect to which is in M. Conditions are specified subject to which the set L(G,H,M) = { (H) | N L_q(M)} forms a lattice under set-theoretic inclusion and the map : L_q(M) L(G,H,M) such that (N) = (H) for any quasivariety N L_q(M)is an antihomomorphism of the lattice L _q (M) onto the lattice L(G, H, M).__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 238–251, March–April, 2005.     

Identities of semigroup algebras of completely O-simple semigroups

Let H = M⁰(G; I, ; P) be a Rees semigroup of matrix type with sandwich matrix P over a group H⁰ with zero. If F is a subgroup of G of finite index and X is a system of representatives of the left cosets of F in G, then with the matrix P there is associated in a natural way a matrix P(F, X) over the group F⁰ with zero. Our main result: the semigroup algebra K[H] of H over a field K of characteristic 0 satisfies an identity if and only if G has an Abelian subgroup F of finite index and, for any X, the matrix P(F, X) has finite determinant rank.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 203–212, August, 1975.     

Groups with Largely Splitting Automorphisms of Orders Three and Four

A subset X of a group G is said to be large (on the left) if, for any finite set of elements g₁,l... ,g_kin G, an intersection of the subsets g_iX=g_imid x in X is not empty, that is, limits_{i=1} ^{k}g_iX . It is proved that a group in which elements of order 3 form a large subset is in fact of exponent 3. This result follows from the more general theorem on groups with a largely splitting automorphism of order 3, thus answering a question posed by Jaber amd Wagner in [1]. For groups with a largely splitting automorphism of order 4, it is shown that if His a normal -invariant soluble subgroup of derived length d then the derived subgroup [H,H] is nilpotent of class bounded in terms of d. The special case where =1 yields the same result for groups that are largely of exponent 4.     

Subgroups in T-groups

A subgroup D of the group G is called polynormal if for it there exists a fan (RZhMat, 1980, 5A208) and all basis subgroups G of this fan are D-complete (i.e., D^G =G). All subgroups in G are polynormal if and only if G is a T-group (the relation of normality for subgroups is transitive).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 62–65, 1980.The authors express gratitude to Z. I. Borevich for posing the problem and interest in the work.     

A note on a result of Skiba

A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H _sG , where H _sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following

Theorem. Let ? be a saturated formation containing the class of all supersoluble groups

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and let G be a group with E a normal subgroup of G such that G/E ∈ ?. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ?.

     

Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = , or of twisted type G = ^m,m = 2,3, over a field K. Its root subgroups X_s, for all possible s G⁺, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U U⁹, g G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U U⁹ in G(K)has a p-multiple index. Put p() = max(r,r)/(s,s) | r,s . We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = and p() is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U of Uⁿ in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and (K,J) be a congruence subgroup of the Chevalley group (K) modulo a nilpotent ideal J. We examine an hypercentral series 1 Z₁ Z₂ ... Z_c-1 of the group U(K) (K,J). Theorem 2 shows that under an extra restriction on the quotient (J_t : J) of ideals, central series are related via Z_i = T_c-iC, 1 i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Z_pm and J = (p^d), 1 d < m, d| m.     

On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period

A proper subgroup H of a group G is said to be strongly isolated if it contains the centralizer of any nonidentity element of H and 2-isolated if the conditions >C _G(g) H 1 and 2(C_G(g)) imply that C_G(g)H. An involution i in a group G is said to be finite if |ii ^g| < (for any g G). In the paper we study a group G with finite involution i and with a 2-isolated locally finite subgroup H containing an involution. It is proved that at least one of the following assertions holds:1) all 2-elements of the group G belong to H;2) (G,H) is a Frobenius pair, H coincides with the centralizer of the only involution in H, and all involutions in G are conjugate;3) G=FFC_G(i) is a locally finite Frobenius group with Abelian kernel F;4) H=V D is a Frobenius group with locally cyclic noninvariant factor D and a strongly isolated kernel V, U=O₂(V) is a Sylow 2-subgroup of the group G, and G is a Z-group of permutations of the set =U ^g g G.     

On the structure of subgroups of a nilpotent nonperiodic group

The structural characteristic of the normal divisor in a locally nilpotent torsion-free group is given. Moreover, a property of structural isomorphisms of locally nilpotent groups containing no less than two independent elements of infinite order is proved: if H is the subgroup of the mentioned group G, N(H) is its normalizer in G, and is a structural isomorphism of the group G, then N(H) = N(H ).Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 389–396, April, 1972.     

An Extension Theorem for Euler Characteristics of Groups

We give a homological definition of the Euler characteristic (G) of a group G; if N is a normal subgroup of G with quotient group H, and if (H) and (N) are defined, then (G) is defined, and is the product of the other two. Several conjectures and problems are proposed.     

On the lattice of subgroups normalized by the symmetric group in a complete monomial group

We consider a lattice of subgroups normalized by the symmetric group S_n in a complete monomial group G = H|S_n, where H is an arbitrary (finite or infinite) group. It is shown that for n3, the subgroup is strongly paranormal in this wreath product for any H. A similar result is obtained for the alternating group A_n, n4. The property of strong paranormality for D in G means that for any element x G, the commutator identity [[x,D],D]=[x, D] holds. This guarantees a standard arrangement of subgroups of G normalized by D. Bibliography: 17 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 111–118.     

Influence of properties of maximal subgroups on the structure of a finite group

We establish some tests for the solvability of finite groups and describe one class of unsolvable groups. We prove that an unsolvable group G such that a maximal subgroup M= P × H is nilpotent and the 2-Sylow subgroup P of M is metacyclic has a normal series GD-g₀TD-{1} such that T is contained in M, G₀/T - PSL (2, q), where q is a power of a prime of the form 2ⁿ± 1 and the index of g₀ in G is not greater than 2.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 183–190, February, 1972.The author warmly thanks V. A. Vedernikov for his guidance and assistance in completing this work.     

Locally nilpotent groups with the min−∞−n condition

The study of locally nilpotent groups with the weak minimality condition for normal subgroups, the min^––n condition, is continued. The following results are obtained.THEOREM 1. A locally nilpotent group with the min^––n condition is countable.THEOREM 2. If G is a locally nilpotent group with the min^––n condition whose periodic part is nilpotent and the orders of the elements of the periodic part are bounded in the aggregate, then G=t(G)A, where the subgroup A is minimax.THEOREM 3. Suppose G is a locally nilpotent group with the min^––n condition and T is its periodic part. If T is nilpotent and G/T is Abelian, then G=TA, where the subgroup A is minimax.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 340–346, March, 1990.     

Quelques proprietes des espaces homogenes spheriques

Let G be a connected, reductive, algebraic group on an algebraically closed field k of characteristic zero. Let H be aspherical subgroup of G, i.e. H is a closed subgroup of G such that every Borel subgroup of G operates on G/H with an open orbit.It is shown that for a spherical subgroup H, the homogeneous space G/H is a deformation of a homogeneous space G/H₀, where H₀ contains a maximal unipotent subgroup of G (such a H₀ is spherical). It is also shown that every Borel subgroup of G has a finite number of orbits in G/H.