Patrizia Longobardi

SMALL DOUBLING IN ORDERED GROUPS

We prove that if

On a combinatorial problem in group theory

\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S

Direct and inverse problems in additive number theory and in non-abelian group theory

is a finite subset of an ordered group that generates a nonabelian ordered group, then

On a Commuting Graph on Conjugacy Classes of Groups

|S^2|\geq 3|S|-2

A small doubling structure theorem in a Baumslag-Solitar group

. This generalizes a classical result from the theory of set addition.

When is a right orderable group locally indicable

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X2|<|X|2, whereX2={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism.

On the structure of subsets of an orderable group with some small doubling properties

Abstract We obtain new direct and inverse results for Minkowski sums of dilates and we apply them to solve certain direct and inverse problems in Baumslag–Solitar groups, assuming appropriate small doubling properties.

Small doubling in ordered groups: Generators and structure

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.

On Groups with Two Isomorphism Classes of Derived Subgroups

We solve a general inverse problem of small doubling type in a monoid, which is a subset of the Baumslag-Solitar group B S ( 1 , 2 ) .

On Infinite Camina Groups

If a group G has an ascending series 1 = G0 ≤ G1 ≤ · · · ≤ Gρ = G of subgroups such that for each ordinal α, Gα C G, and Gα+1/Gα has no nonabelian free subsemigroup, then G is right orderable if and only if it is locally indicable. In particular if G is a radical-by-periodic group, then it is right orderable if and only if it is locally indicable.