Let Aut(X, B) be the group of all Borel automorphisms of a standard Borel space (X, B). We study topological properties of Aut(X, B) with respect to the uniform and weak topologies, τ and p, defined in [BDK1]. It is proved that the class of smooth automorphisms is dense in (Aut(X, B), p). Let Ctbl(X) denote the group of Borel automorphisms with countable support. It is shown that the topologica...

We describe some of the geometric properties of the automorphism group Aut(F ) of Thompson’s group F . We give realizations of Aut(F ) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F ) and their metric properties. In particular, we show that the subg...

If (W, S) is a right-angled Coxeter group, then Aut(W ) is a semidirect product of the group Aut◦(W ) of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, Aut◦(W ) is a semidirect product of Inn(W ) by the quotient Out◦(W ) = Aut◦(W )/Inn(W ). We also give sufficient conditions for the compatibility of the two semidirect products. When...

The present study examined the verbal memory profile and its relation to organizational strategies in high-functioning (Hi-AUT) and low-functioning (Lo-AUT) children with autism. Twenty-two Hi-AUT and 16 Lo-AUT, and 22 age-, genderand handednessmatched normal children (NC) were required to remember a list of semantically related words for immediate and delayed recall and recognition. All autist...

We study the automorphisms of a graph product of finitely-generated abelian groups W . More precisely, we study a natural subgroup Aut W of AutW , with Aut W = AutW whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition Aut W = (InnW ⋊Out W )⋊Aut W . We also give a number of applications, some of wh...

We define a four-dimensional Lie algebra g in this paper and then prove that is solvable but not nilpotent. Due to the fact algebra, ∀x,y∈g,[x,y]=−[y,x], is, operation [,] has anti symmetry. Symmetry very important law, antisymmetry also law. studied structure of Poisson algebras on using matrix method. necessary sufficient conditions for automorphism class algebras, give decomposition its grou...

The Cayley graphs Gm(G)=Cay(Gm,S) of Cartesian powers the finite group G with respect to canonical symmetric subsets S=S(m)⊂Gm are investigated. It is proved that for each m>1 graph Gm(G) determines up isomorphism. groups automorphisms Aut(Gm(G)) determined. shown if a non-abelian group, then Aut(Gm(G))≃(Gm⋊Aut(G))⋊Dm+1, where Dm+1 dihedral order 2m+2. If an abelian (with some exceptions m=3), ...

Let G = (V,E) be a simple graph with exactly n vertices and m edges. The aim of this paper is a new method for investigating nontriviality of the automorphism group of graphs. To do this, we prove that if |E| >=[(n - 1)2/2] then |Aut(G)|>1 and |Aut(G)| is even number.