Let X be a proper hyperbolic geodesic metric space of bounded growth and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not amenable then the second continuous bounded cohomol-ogy group H 2 cb (G, L 2 (G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ [1, ∞) the second continuous bounded cohomology group H 2 cb (G, L p (G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).

In this paper, the definition of ?-interval valued fuzzy subgroup is introduced by help set which constructed on whose elements are closed sub-intervals including ? unit interval I = [0, 1]. The fundamental and structural properties these groups examined. Some definitions, propositions examples about given.

The purpose of this paper is to introduce the concept of pairwise F-closedness in bitopological spaces. This space contains both of pairwise strongcompactness and pairwise S-closedness and contained in pairwise quasi H-closedness. The characteristics and relationships concerning this new class of spaces with other corresponding types are established. Moreover, several of its basic and important...

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight “forbidden” graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or co...

(4.) Note that in Theorem 5.4, the case where A is trivial [i.e., is equal to the anabelioid associated to the trivial group {1}] is not excluded. Thus, suppose that, in Theorem 5.4, we assume further that A is trivial. Then let us observe that this implies that the underlying graph of G [orH] consists of a single vertex and no edges. [Indeed, if the underlying graph of G has at least one edge,...

Let [Formula: see text] be a finite group and fixed prime divisor of text]. We prove that if every maximal subgroup is nilpotent, or normal, has text]-order, then (1) solvable; (2) Sylow tower; (3) there exists at most one such neither text]-nilpotent nor text]-closed.

Determining the subgroup structure of algebraic groups (over an algebraically closed field K of arbitrary characteristic) often requires an understanding of those instances when a group Y and a closed subgroup G both act irreducibly on some module V , which is rational for G and Y . In this paper and in Overgroups of irreducible linear groups, I (J. Algebra 181 (1996), 26–69), we give a classif...

Let G be a linear algebraic group, H a closed subgroup of G and X a G-variety. This paper explores the connection between G-orbits and H-orbits in X, concentrating in particular on the question of when we have the implications G · x closed in X implies H · x closed in X, and vice versa. Some of the general results found in this paper are then used in the special case where G and H are reductive...