Valentina Grazian, Carmine Monetta,
In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if G and H are finite groups with isomorphic non-commuting graphs and G is nilpotent, then H must be nilpotent as well (Conjecture 2). We characterize the structure of such an H when G is a finite AC-group, that is, a finite group in which all centralizers of non-central elements are abelian. As an application, we prove Conjecture 2 ...
Tópico(s): graph theory and CDMA systems
2023 - Elsevier BV | Journal of Algebra
Valentina Grazian, Carmine Monetta, Marialaura Noce,
In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of p′-order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors.
Tópico(s): Coding theory and cryptography
2024 - Taylor & Francis | Communications in Algebra
Produção Nacional
Revisado por pares
Yerko Contreras-Rojas, Valentina Grazian, Carmine Monetta,
In this paper we address the problem of understanding when a verbal subgroup of a finite group is p-nilpotent, with p a prime, that is, when all its elements of p′-order determine a subgroup. We provide two p-nilpotency criteria, one for the terms of the lower central series of any finite group and one for the terms of the derived series of any finite soluble group, which relies on arithmetic properties related to the order of products of commutators.
Tópico(s): semigroups and automata theory
2022 - Elsevier BV | Journal of Algebra