Giovanni Vincenzi

ON GROUPS WHICH CONTAIN NO HNN-EXTENSIONS

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin.

Fibonacci-like sequences and generalized Pascal’s triangles

The properties pertaining to diagonals of generalized Pascal’s triangles are studied. Combinatorial relationships between Fibonacci-like sequences and Fibonacci sequence itself are determined, using the sequence of diagonals of generalized Pascal’s triangle.

GROUPS WHOSE PROPER SUBGROUPS ARE GENERALIZED FC-GROUPS

Let 𝔛 be a class of groups. A group G is said to be minimal non-𝔛 if all proper subgroups of G are 𝔛-groups but G itself is not. The aim of this paper is to study the class of minimal non-FCn-groups, where FCn (n is a positive integer) is a class of generalized FC-groups introduced in [F. de Giovanni, A. Russo and G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J.28 (2002) 241–254].

Tribonacci-like sequences and generalized Pascal's pyramids

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascals pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinbergs triangle associated to a suitable generalized Pascals pyramid. The results also extend similar properties of Fibonacci-like sequences.

Groups without proper abnormal subgroups

Abstract A subgroup H of a group G is said to be abnormal in G if g ∈ for each element g ∈ G. It is well known that every locally nilpotent group has no proper abnormal subgroups, but it is an open question whether the converse holds. In this article we prove this conjecture for some classes of infinite groups. In particular, it is proved that an FC-nilpotent group without proper abnormal subgroups is hypercentral. Also groups with finitely many abnormal subgroups are considered.

Groups in which normality is a weakly transitive relation

We study groups in which normality is a weakly transitive relation, giving an extension of in Theorem A [On finite T-groups, J. Aust. Math. Soc. 75 (2003) 181–191] due to Ballester-Bolinches and Esteban-Romero pointing out the relations between these groups and those in which all subgroups are almost pronormal. Moreover, we extend a well-known theorem of Peng [Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc. 20 (1969) 232–234] proving that for a large class of generalized FC-groups the weak transitivity of normality is equivalent to having finitely many maximal pronormalizers of subgroups.

On the Wielandt subgroup of generalized FC-groups

We extend to soluble FC*-groups, the class of generalized FC-groups introduced in [de Giovanni, Russo and Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28(3) (2002) 241–254], the characterization of finite soluble T-groups, and some results on the Wielandt subgroup, obtained recently in [Kaplan, On finite T-groups and the Wielandt subgroup, J. Group Theory. 14 (2011) 855–863].

From Fibonacci Sequence to the Golden Ratio

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

Groups whose all subgroups are ascendant or self-normalizing

This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16].

Logarithmic spirals and continue triangles

In this article we will use some special triangles, to construct polygonal chains that describe the families of logarithmic spirals, among which are the celebrated Golden Spiral, Spira solaris and Pheidia Spiral.