On Christoffel and standard words and their derivatives

We introduce and study natural derivatives for Christoffel and finite standard words, as well as for characteristic Sturmian words. These derivatives, which are realized as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp.,~$ab^{k}$) and $b$ to $a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are longer than one letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it.Comment: 28 pages. Final version, to appear in TC

Dualpronormality and Fitting classes

In this paper we extend dualpronormality to an arbitrary Fitting class, introducing the notion of F-dualpronormality, and investigate groups having a structure of F-dualpronormal subgroups which is particularly rich or particularly poor

Pronormalità duale e normalità nella classe dei gruppi finiti risolubili

In this paper we consider a subgroup embedding property weaker than normality, dualpronormality. We see that, if "many" subgroups are dualpronormal, then they are all normal, and we get new characterizations of t-groups

On normal Fitting classes

The paper contains some characterizations of normal Fitting classes

Sull'esistenza di sottogruppi nilpotenti autonormalizzanti in alcuni gruppi semplici, II

We prove that in the Mathieu groups there is a unique conjugacy class of nilpotent self-normalizing subgroups, the class of the 2-Sylow subgroups. In the Janko group J1 there are no nilpotent self-normalizing subgroups

Alcune considerazioni aritmetiche sull'ordine dei gruppi semplici di Suzuki

It is proved that within the class of finite simple groups, the Suzuki groups are uniquely determined by the set of prime divisors of their order

Su una classe di gruppi finiti supersolubili

In this paper we study the class of finite groups G whose nilpotent residual is a Hall subgroup having all subgroups normal in G

Su alcune classi di sottogruppi pronormali duali

In this paper we introduce the concept of dualpronormal subgroup and we use it to study groups in which Carter subgroups and N-injectors coincide

Prodotti degli elementi di un gruppo finito

We consider the subset Pr(G) of a finite group G whose elements are product of all elements of G, in any order, and characterize nilpotent groups in terms of Pr(G)