let $g$ be a group and $aut(g)$ be the group of automorphisms of‎‎$g$‎. ‎for any natural‎‎number $m$‎, ‎the $m^{th}$-autocommutator subgroup of $g$ is defined‎‎as‎: ‎$$k_{m}(g)=langle[g,alpha_{1},ldots,alpha_{m}] |gin g‎,‎alpha_{1},ldots,alpha_{m}in aut(g)rangle.$$‎‎in this paper‎, ‎we obtain the $m^{th}$-autocommutator subgroup of‎‎all finite abelian groups‎.

Let D be a symmetric 2-(121, 16, 2) design with the automorphism group of Aut(D). In this paper the order of automorphism of prime order of Aut(D) is studied, and some results are obtained about the number of fixed points of these automorphisms. Also we will show that |Aut(D)|=2p 3q 5r 7s 11t 13u, where p, q, r, s, t and u are non-negative integers such that r, s, t, u ? 1. In addition we prese...

abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

We prove that the isomorphism of scattered tree automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders are undecidable. For the existence of automatic automorphisms of word automatic linear orders, we determine the exact level of undecidability in the arithmetical hierarchy.

We give a necessary and sufficient condition for k-step nilmanifolds associated with graphs (k ≥ 3) to admit Anosov automorphisms. We also prove nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds. 2000 Mathematics Subject Classification. Primary 37D20; Secondary 22E25

The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.

In the paper we study the endomorphim semigroup of a general quantum polynomial ring, its finite groups of automorphisms and homological properties of this ring as a module over the skew group ring of a finite group of automorphisms. Moreover properties of the division ring of fractions are considered.

We study whether the basin of attraction of a sequence of automorphisms of C is biholomorphic to C. In particular we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to C if the maps are repeated often enough. We also construct Fatou-Bieberbach domains whose boundaries are 4-dimensional.

We classify certain automorphisms of an even unimodular lattice L with fixed irreducible and unramified characteristic polynomial. The question which automorphisms are conjugate to their inverses in the orthogonal group of L is investigated. 2000 Math. Subj. Class. 11H56, 11R06, 20F55.

Imposing a group structure on the set of random orders, the paper reformulates and characterizes the random order value and more generally semi-value operators in a uni ed framework that encompasses games with nite and continuum of players and allows symmetry of the operators to be with respect to a subgroup of automorphisms. A set of orderings of players equipped with a group structure induced...