Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

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].     

Vertex-transitive graphs: Symmetric graphs of prime valency

Let G be a group acting symmetrically on a graph Σ, let G₁ be a subgroup of G minimal among those that act symmetrically on Σ, and let G₂ be a subgroup of G₁ maximal among those normal subgroups of G₁ which contain no member except 1 which fixes a vertex of Σ. The most precise result of this paper is that if Σ has prime valency p, then either Σ is a bipartite graph or G₂ acts regularly on Σ or G₁ | G₂ is a simple group which acts symmetrically on a graph of valency p which can be constructed from Σ and does not have more vertices than Σ. The results on vertex-transitive groups necessary to establish results like this are also included.     

Locally minimal topological groups 1

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G,τ) is called locally minimal if there exists a neighborhood U of 0 in τ such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than τ. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. Unlike the minimal topologies, the locally minimal ones are always available on “large” groups. To support this line, we prove that a bounded abelian group G admits a non-discrete locally minimal and locally GTG group topology iff |G|?c.     

Locally minimal topological groups 2

We continue in this paper the study of locally minimal groups started in Außenhofer et al. (2010) [4]. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group G almost minimal if it has a closed, minimal normal subgroup N such that the quotient group G/N is uniformly free from small subgroups. The class of almost minimal groups includes all locally compact groups, and is contained in the class of locally minimal groups. On the other hand, we provide examples of countable precompact metrizable locally minimal groups which are not almost minimal. Some other significant properties of this new class are obtained.     

Groups with few conjugacy classes of non-normal subgroups

The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.      

Uniform structures and square roots in topological groups

Topological groups which are free from small subgroups and topological groups with locally uniformly continuous group multiplication are considered. Results concerning square roots, one-parameter subgroups and extensions of local groups are obtained as well as some generalisations of theorems for locally compact groups.     

The action of nilpotent groups on infinite graphs

First we prove a result about the action of nilpotent groups on the set of ends of locally finite graphs. This theorem has immediate consequences for the structure of graphs which allow a transitive action of those groups. Further we investigate the cycle-structure of automorphisms of a transitive nilpotent group and the existence of abelian groups acting on sets of imprimitivity of graphs whose automorphism groups have transitive nilpotent subgroups.     

Groups with finiteness conditions on the lower central series of non-normal subgroups

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group G the subgroup \(\gamma _{k}(G)\) is finite if the set \(\{\gamma _{k}(H)\;|\;H\le G,\,H\ntriangleleft G\}\) is finite. Moreover, locally graded groups with finitely many kth terms of lower central series of infinite non-normal subgroups are also completely described.     

Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

Groups with many subgroups having a transitive normality relation

A group is said to be aT-group if all its subnormal subgroups are normal. The structure of groups satisfying the minimal condition on subgroups that do not have the propertyT is investigated. Moreover, locally soluble groups with finitely many conjugacy classes of subgroups which are notT-groups are characterized.     

Bisimple inverse semigroups whose idempotents are an interval

Bisimple locally compact topological inverse semigroups whose maximal subgroups are compact and whose set of idempotents is a real interval are studied in this paper. The structure of such semigroups is shown to be entirely determined by their maximal subgroups in case the semigroups come from a certain subclass which includes the connected ones. The factor semigroups moduloH are completely determined.     

An algebraic structure for Faber polynomials

Let Σ be the set of functions, convergent for all |z|>1, with a Laurent series of the form f(z)=z+∑_n?0a_nz^-n. In this paper, we prove that the set of Faber polynomial sequences over Σ and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the Bell^(k) subgroup of the Riordan group, respectively. Further, a relationship between such Faber polynomial sequences and Lucas and Sheffer polynomial sequences is derived.     

Controllability in codimension one

Local and global controlability of analytic affine control systems Σ with an arbitrary number of controls are studied, assuming strong accessibility of Σ and codimension one of the Lie algebra T′ generated by the input vector fields, i.e., dim T′ (p) = n − 1 at every p ϵ M. Controls are assumed to have no a priori bound. From the study of the set H where a necessary condition for local controllability at a point is verified and assuming some transversality relations, an easy to verify geometric condition is proved: H is a submanifold and Σ is locally controllable at every point of H outside a codimension one submanifold. A geometric sufficient condition for global controllability on simply connected manifolds is then obtained: if every leaf of T′ intersects the manifold H and some transversality relations (including those involved in the local condition) are verified, Σ is globally controllable.