Cyclotomic Polynomials and Finite Automorphisms of Groups

We introduce minimal polynomials for finite automorphisms of commutative groups and relate them to the exponent of the fixed points and to the reducibility of the group. Some results can be extended to the noncommutative case.     

Automorphisms of Free Groups and Outer Space

This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations.     

Automorphisms of the Two Dimensional Crystallographic Groups

We compute the groups Aut(G) and Out(G) where G is a crystallographic group of rank 2, or so-called wallpaper group.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

The structure of transitive ordered permutation groups

We give some necessary and sufficient conditions for transitive l-permutation groups to be 2-transitive. We also discuss primitive components and give necessary and sufficient conditions for transitive l-permutation groups to be normal-valued.     

Metabelian Products of Groups

We prove a number of facts on metabelian products of metabelian groups, useful in algebraic geometry over groups. Namely, for a metabelian product of arbitrary metabelian groups, we look at the structure of a derived subgroup, and the Fitting radical; find criteria determining when a metabelian product of u-groups is again a u-group; and specify conditions under which a metabelian product of metabelian groups is a strong semidomain.     

Distinguished extensions of a lattice-ordered group

Any lattice-ordered group (l-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, anl-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on thel-group must be injective on the extension as well. Thus nol-group has a maximal essential extension in the categoryIGp ofl-groups withl-homomorphisms. However, anl-group is a distributive lattice, and so has a maximal essential extension in the categoryD of distributive lattices with lattice homomorphisms. Adistinguished extension of onel-group by another is one which is essential inD. We characterize such extensions, and show that everyl-groupG has a maximal distinguished extensionE(G) which is unique up to anl-isomorphism overG.E(G) contains most other known completions in whichG is order dense, and has mostl-group completeness properties as a result. Finally, we show that ifG is projectable then E(G) is the -completion of the projectable hull ofG.Presented by M. Henriksen.     

Groups whose non-subnormal subgroups have a transitive normality relation

This article investigates the structure of groups in which every subgroup either is subnormal or has a transitive normality relation, with special attention to the case in which subnormal subgroups have bounded defect.     

Noetherian Automorphisms of Groups

An automorphism α of a group G is called a noetherian automorphism if for each ascending chain

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve then either or . In the former case is projectively equivalent to the curve with equation X^q+1Y+Y^q+1+X=0 studied by Pellikaan. Furthermore, the curve has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its -rational points can be partitioned into finite projective planes . In this paper, the full automorphism group of such curves is determined. It turns out that is the normalizer of a Singer group in .     

Rigidity of Branch Groups Acting on Rooted Trees

Automorphisms of groups acting faithfully on rooted trees are studied. We find conditions under which every automorphism of such a group is induced by a conjugation from the full automorphism group of the rooted tree. These results are applied to known examples such as Grigorchuk groups, Gupta–Sidki group, etc.     

Contractive Automorphisms of Locally Compact Groups and the Concentration Function Problem

Let G be a noncompact locally compact group. We show that a necessary and sufficient condition in order that G support an adapted probability measure whose concentration functions fail converge to zero is that G be the semidirect product , where is an automorphism of N contractive modulo a compact subgroup. Any adapted a probability measure whose concentration functions fail to converge to zero has the form =v×₁ where v is a probability measure on N. If G is unimodular then the concentration functions of an adapted probability measure fail to converge to zero if and only if is supported on a coset of a compact normal subgroup.     

Monochromatic Paths and at Most 2-Coloured Arc Sets in Edge-Coloured Tournaments

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m-colours. If v is a vertex of an m-coloured tournament T, we denote by ξ(v) the set of colours assigned to the arcs with v as an endpoint. In this paper is proved that if T is an m-coloured tournament with |ξ(v)|≤2 for each vertex v of T, and T satisfies at least one of the two following properties (1) m≠3 or (2) m=3 and T contains no C₃ (the directed cycle of length 3 whose arcs are coloured with three distinct colours). Then there is a vertex v of T such that for every other vertex x of T, there is a monochromatic directed path from x to v. Received: April, 2003     

Central Automorphisms and Inner Automorphisms in Finitely Generated Groups

Let G be a group and Aut_c(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Aut_c(G) = Inn(G) if and only if Z(G) = G′ and Z(G) is cyclic. But we shown that we cannot extend this result for infinite groups. In fact, there exist finitely generated nilpotent groups of class 2 in which G′ =Z(G) is infinite cyclic and Inn(G) < C* = Aut_c(G). In this article, we characterize all finitely generated groups G for which the equality Aut_c(G) = Inn(G) holds.     

有限ATI-群的类保持Coleman自同构

设G是一个有限群,对G的任意阿贝尔子群A及任意g∈G,若A∩A~g=1或A,则称G为一个ATI-群.本文证明了,对任意p∈τ(G),如果ATI-群G的一个p-方幂阶类保持自同构在G的任意Sylow子群上的限制等于G的某个内自同构的限制,则它必定是一个内自同构.作为该结果的一个直接推论,我们也证明了有限ATI-群G有正规化性质.     

Some remarks on almost l-groups

The divisibility group of every Bézout domain is an abelian l-group. Conversely, Jaffard, Kaplansky, and Ohm proved that each abelian l-group can be obtained in this way, which generalizes Krull’s theorem for abelian linearly ordered groups. Dumitrescu, Lequain, Mott, and Zafrullah [3] proved that an integral domain is almost GCD if and only if its divisibility group is an almost l-group. Then they asked whether the Krull-Jaffard-Kaplansky-Ohm theorem on l-groups can be extended to the framework of almost l-groups, and asked under what conditions an almost l-group is lattice-ordered [3, Questions 1 and 2]. This note answers the two questions. Received: 29 April 2008