Mercede Maj

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

On the derived length of groups with some permutational property

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

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

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

On a Commuting Graph on Conjugacy Classes of Groups

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

A small doubling structure theorem in a Baumslag-Solitar group

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

Define P=Una2Pn and Q=Unr2Q,,. Obviously P,sQ,, so PEQ. The class P was first introduced in [lo], in the context of semigroups. Many authors have studied groups in P or in Q. It has been shown that P is the class of finite-by-abelian-by-finite groups (see [S]) and that this is also the class Q (see [2]). Hence P = Q. Obviously P, = Q2 is the class of abelian groups, while, for every n > 2, P, is always a proper subclass of Qn (see [2]). The class P, is the class of groups with derived subgroup of order at most 2 (see [4]), and groups in P4 are metabelian (see [7]), while groups in P5 or P, are soluble (see [3]). Also known is a complete description of the class P, (see El, 681). In [2] R. D. Blyth and D. J. S. Robinson proved that a group in Q4 is always soluble and asked if a group in Q3 is metabelian. In this paper we give an affirmative answer to this question (see Section 2). Moreover we prove, more generally that the derived length I(G)

Small doubling in ordered groups: Generators and structure

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.

On Groups with Two Isomorphism Classes of Derived Subgroups

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 Infinite Camina Groups

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 ) .