Nicola Grittini

University of Florence | UNIFI · Dipartimento di Matematica e Informatica "Ulisse Dini"

We prove that, in a finite group, if every rational irreducible character has odd degree, then all rational elements are 2-elements, as it was originally conjectured by Tiep and Tong-Viet.

In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.

In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.

It is known that, if all the irreducible real valued characters of a finite group are of odd degree, then the group has a normal Sylow 2-subgroup. In this paper, we prove and analogous result for solvable groups, by taking into account the degree of irreducible characters fixed by some field isomorphism of prime order $p$. We prove it as a conseque...

If a group G is π-separable, where π is a set of primes, the set of irreducible characters B π ⁡ ( G ) ∪ B π ′ ⁡ ( G ) {\operatorname{B}_{\pi}(G)\cup\operatorname{B}_{\pi^{\prime}}(G)} can be defined. In this paper, we prove variants of some classical theorems in character theory, namely the theorem of Ito–Michler and Thompson’s theorem on characte...

If a group $G$ is $\pi$-separable, where $\pi$ is a set of primes, the set of irreducible characters $\operatorname{B}_{\pi}(G) \cup \operatorname{B}_{\pi'}(G)$ can be defined. In this paper, we prove that there are variants of some classical theorems in character theory, namely the Theorem of Ito-Michler and Thompson theorem on character degrees,...

Let G be a p-solvable group, where p is a prime. We prove that the p-length of G is less or equal then the number of distinct irreducible character degrees of G not divisible by p. Furthermore, we prove that the result still holds if we impose some restriction on the field of values of the characters. In particular, if p=2, we can consider only rat...

Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that χ(g) ≠ 0 for all irreducible characters χ of G. Such an element is said to be non-vanishing in G. Let p be a prime. If all p-elements of N satisfy the previous property, then we prove that N has a normal Sylow p-subgroup. As a consequence, we al...

Let N be a normal subgroup of a finite group G. An element g ∈ G such that χ(g) = 0 for some irreducible character χ of G is called a vanishing element of G. The aim of this paper is to analyse the influence of the π-elements in N which are (non-)vanishing in G on the π-structure of N , for a set of primes π. We also study certain arithmetical prop...