In this paper, we study the complete ⋆-metric semigroups and groups Raǐkov completion of invariant groups. We obtain following. (1) Let (X,d⋆) be a space containing semigroup (group) G that is dense subset X. If restriction d⋆ on invariant, then X can become as subgroup, (2) (G,d⋆) group such G. Then, if only (G,τd⋆) complete.

Let X be a topological abelian group and H subgroup of X. We find conditions under which local quasi-convexity both X/H results in the same property for This is true instance if precompact, or metrizable dually embedded also either discrete bounded torsion. give some general principles point out errors we have found existing literature on this problem.

Let X be a normal projective algebraic variety, G its largest connected automorphism group, and A(G) the Albanese variety of G . We determine the isogeny class of A(G) in terms of the geometry of X . In characteristic 0, we show that the dimension of A(G) is the rank of every maximal trivial direct summand of the tangent sheaf of X . Also, we obtain an optimal bound for the dimension of the lar...

Let G be a group. Then subgroup A of is said to modular in if (i) $$\langle X, \cap Z \rangle =\langle Z$$ for all $$X \le G, G$$ such that , and (ii) A, Y $$Y $$A . We obtain description finite groups which modularity transitive relation.

We show for a finite abelian group G and any element in the image of the Swan homomorphism sw : Z/|G|∗ −→ K̃0(ZG) that it can be realized as the finiteness obstruction of a finitely dominated connected CW -complex X with fundamental group π1(X) = G such that π1(X) is equal to the subgroup G1(X) defined by Gottlieb. This is motivated by the observation that any H-space X satisfies π1(X) = G1(X) a...

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.

A Polish group G is called a group of quasi-invariance or a QI-group, if there exist a locally compact group X and a probability measure μ on X such that 1) there exists a continuous monomorphism of G to X, and 2) for each g ∈ X either g ∈ G and the shift μg is equivalent to μ or g 6∈ G and μg is orthogonal to μ. It is proved that G is a σ-compact subset of X. We show that there exists a quotie...

Let n be a positive integer or infinity (denote ∞). We denote by W ∗(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X0 ⊆ X, with 2 ≤ |X0| ≤ n + 1 and a function f : {0, 1, 2, . . . , k} −→ X0, with f(0) 6= f(1) and non-zero integers t0, t1, . . . , tk such that [x0 0 , x t1 1 , . . . , x tk k ] = 1, where xi := f(...

Let I = (a,b) and L be a nowhere dense perfect set containing the ends of the interval I and let φ : I → R be a non-increasing continuous surjection constant on the components of I \ L and the closures of these components be the maximal intervals of constancy of . The family F t t R of the interval-valued functions F t 1 φ { , ∈ } (x): = φ [t + φ(x)], x ∈ I forms a set-valued iteration group. W...