On Complementability of Nonmetacyclic Subgroups in Finite A-Groups

We study groups G that satisfy the following conditions: (i) G is a finite solvable group with nonprimary metacyclic second subgroup and (ii) all Sylow subgroups of the group G are elementary Abelian subgroups. We describe the structure of groups of this type with complementable nonmetacyclic subgroups.     

Finite 2-groups with exactly one nonmetacyclic maximal subgroup

We determine here the structure of the title groups. All such groups G will be given in terms of generators and relations, and many important subgroups of these groups will be described. Let d(G) be the minimal number of generators of G. We have here d(G) ≤ 3 and if d(G) = 3, then G′ is elementary abelian of order at most 4. Suppose d(G) = 2. Then G′ is abelian of rank ≤ 2 and G/G′ is abelian of type (2, 2^m), m ≥ 2. If G′ has no cyclic subgroup of index 2, then m = 2. If G′ is noncyclic and G/Φ(G ₀) has no normal elementary abelian subgroup of order 8, then G′ has a cyclic subgroup of index 2 and m = 2. But the most important result is that for all such groups (with d(G) = 2) we have G = AB, for suitable cyclic subgroups A and B. Conversely, if G = AB is a finite nonmetacyclic 2-group, where A and B are cyclic, then G has exactly one nonmetacyclic maximal subgroup. Hence, in this paper the nonmetacyclic 2-groups which are products of two cyclic subgroups are completely determined. This solves a long-standing problem studied from 1953 to 1956 by B. Huppert, N. Itô and A. Ohara. Note that if G = AB is a finite p-group, p > 2, where A and B are cyclic, then G is necessarily metacyclic (Huppert [4]). Hence, we have solved here problem Nr. 776 from Berkovich [1].     

Finite A-groups with complementable nonmetacyclic subgroups

We study groups G that satisfy the following conditions (i) G is a finite solvable group with nonidentity primary metacyclic second commutator subgroup (ii) all Sylow subgroups of G are elementary Abelian. We describe the structure of groups of this type with complementable nonmetacyclic subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 607–615, May, 2006.     

Finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions

We classify here nonmetacyclic finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions (Theorem 1.1). This solves problem Nr. 710 for p = 2 stated by Y. Berkovich in [1]. For p > 2 the corresponding problem is open. Received: 14 March 2007 Revised: 15 April 2007     

On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

Groups with elementary abelian commutant of at most p 2th order

We obtain a representation of nilpotent groups with a commutant of the type (p) or (p, p) that has the form of a product of two normal subgroups. One of these subgroups is constructively described as a Chernikov p-group of rank 1 or 2, and the other subgroup has a certain standard form. We also obtain a representation of nonnilpotent groups with a commutant of the type (p) or (p, p) in the form of a semidirect product of a normal subgroup of the type (p) or (p, p) and a nilpotent subgroup with a commutant of order p or 1. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 534–539, April, 1998.     

G-covering systems of subgroups for the class of supersoluble groups

Let ? be a class of groups. Given a group G, assign to G some set of its subgroups Σ = Σ(G). We say that Σ is a G-covering system of subgroups for ? (or, in other words, an ?-covering system of subgroups in G) if G ∈ ? wherever either Σ = ? or Σ ≠ ? and every subgroup in Σ belongs to ?. In this paper, we provide some nontrivial sets of subgroups of a finite group G which are G-covering subgroup systems for the class of supersoluble groups. These are the generalizations of some recent results, such as in [1–3].     

On subgroups of finite <Emphasis Type="Italic">p</Emphasis>-groups

In §2, we prove that if a 2-group G and all its nonabelian maximal sub-groups are two-generator, then G is either metacyclic or minimal non-abelian. In §3, we consider a similar question for p > 2. In §4 the 2-groups all of whose minimal nonabelian subgroups have order 16 and a cyclic subgroup of index 2, are classified. It is proved, in §5, that if G is a nonmetacyclic two-generator 2-group and A, B, C are all its maximal subgroups with d(A) ≤ d(B) ≤ d(C), then d(C) = 3 and either d(A) = d(B) = 3 (this occurs if and only if G/G′ has no cyclic subgroup of index 2) or else d(A) = d(B) = 2. Some information on the last case is obtained in Theorem 5.3.     

Some cases of preservation of the Pontryagin dual by taking dense subgroups

We study the compact-open topology on the character group of dense subgroups of topological abelian groups. Permanence properties concerning open subgroups, quotients and products are considered. We also present some representative examples. We prove that every compact abelian group G with w(G)?c has a dense pseudocompact group which does not determine G; this provides (under CH) a negative answer to a question posed by S. Hernández, S. Macario and the third listed author two years ago.     

Generalization of Berg-Dimovski convolution in spaces of analytic functions

In the space H(G) of functions analytic in a ρ-convex region G equipped with the topology of compact convergence, we construct a convolution for the operator J _π+L where J _ρ is the generalized Gel’fond-Leont’ev integration operator and L is a linear continuous functional on H(G). This convolution is a generalization of the well-known Berg-Dimovski convolution. We describe the commutant of the operator J _π+L in ?(G) and obtain the representation of the coefficient multipliers of expansions of analytic functions in the system of Mittag-Leffler functions.     

Group-theoretic compactification of Bruhat-Tits buildings

Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables us to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the Bruhat-Tits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups and the other is about subgroups with distal adjoint action.     

On positive and constructive groups

Considering a group with unique roots (i.e., an R-group), we give a sufficient condition for the existence of a positive (constructive) enumeration with respect to which the isolator of the commutant is computable. Basing on it, we prove the constructivizability of an R-group that admitting a positive enumeration for which the dimension of the commutant is finite. We obtain a necessary and sufficient condition of constructivizability for a torsion-free nilpotent group for which the dimension of the commutant is finite.     

On the Steinhaus property in topological groups

Let G be a locally compact Abelian group and μ a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ₀?m?|G| there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erd?s-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits ²|G|-many μ-nonmeasurable dense subgroups.     

Procyclic coverings of commutators in profinite groups

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G′ such that the order of M is m-bounded and G′/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G′ is either finite of m-bounded order or procyclic.     

Flows, fixed points and rigidity for Kleinian groups

The main application of the techniques developed in this paper is to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension at least three) for G ₁, G ₂ cocompact Kleinian groups, any quasiconformal map pairing a G ₁-invariant pattern to a G ₂-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas and Mj (Pattern rigidity in hyperbolic spaces: duality and pd subgroups, arxiv:math.GT/08094449, 2008) who proved rigidity for Poincare Duality subgroups. Pattern rigidity is a consequence of the study conducted in this paper of the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G ₁ and a quasiconformal conjugate h ^?1 G ₂ h of a cocompact group G ₂. We show that if the conjugacy h is not conformal then this group contains a flow, i.e. a non-trivial one parameter subgroup. Mostow rigidity is an immediate consequence.     

Finite groups with sn-embedded or s-embedded subgroups

A number of authors have studied the structure of a finite group G under the assumption that some subgroups of?G are well located in?G. We will generalize the notion of s-permutable and s-permutably embedded subgroups and we will obtain new criterions of p-nilpotency and supersolvability of groups. We also generalize some known results.     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.