Dual-standard subgroups of finite and locally finite groups

A subgroup D of a group G is called dual-standard if, for all subgroups X and Y of G, <XχD,YχD>=<X,Y>χD. When G is finite, Zappa has given some information concerning the way in which D is embedded in G and the structure of G itself. Among other things, Zappa makes reference to the maximal normal Hall subgroup L of G contained in D. In general L can be arbitrary. The main result of this work, however, is to show that the commutator of L with a complement in G is nilpotent.     

Uniformly recurrent subgroups of simple locally finite groups

It is shown that for the conjugation action of the symmetric group Sn, when n = 6 or n ≥ 8, all S _n -irreducibles appear as constituents of a single conjugacy class, namely, one indexed by a partition λ of n with at least two parts, whose parts are all distinct and taken from the set of odd primes and 1. A simple characterisation of conjugacy classes containing all irreducibles is proved.     

Locally normal subgroups of simple locally compact groups

We announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with open normaliser, and show that its properties reflect the global structure of the ambient group.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Centralizers of finite nilpotent subgroups in locally finite groups

We investigate the structure of locally finite groups with a finite subgroup whose centralizer is close to a linear group. Deceased. Translated fromAlgebra i Logika, Vol. 35 No. 4, pp. 389–410, July–August, 1996.     

On groups factorizable in commuting almost locally normal subgroups

We prove that anRN-group (in particular, locally solvable)G =G ₁ G ₂ ...G _n withG _i and π(G _i ) ∩ π(G _j ) = ?,i ≠j is a periodic hyper-Abelian group if the subgroupsG _j are almost locally normal.     

A note on weakly compact subgroups of locally quasi-convex groups

In this note two results about compact subgroups of locally quasi-convex groups are shown:

1. The quotient group of a locally quasi convex Hausdorff group modulo a compact subgroup is again locally quasi-convex.
2. If a subgroup of a locally quasi-convex group is compact in the weak topology, it is also compact in the original topology.

     

Discrete subgroups of algebraic groups over local fields of positive characteristics

It is shown in this paper that ifG is the group ofk-points of a semisimple algebraic groupG over a local fieldk of positive characteristic such that all itsk-simple factors are ofk-rank 1 and Γ ⊂G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that ifG has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.     

On the existence of stably nilpotent subgroups in locally nilpotent groups

A condition for the existence of stably nilpotent subgroups of a given nilpotency class in a locally nilpotent group is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 361–368, September, 1968.     

Extension of positive definite functions from subgroups of nilpotent locally compact groups

We investigate the problem of when closed subgroups of a nilpotent locally compact group have the property that every continuous positive definite function on extends to some such function on .