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Silvio Dolfi

Publications and source records attributed to Silvio Dolfi.

At least 19 recordsLinked to original sources
arXiv2026

Degrees of $p$-rational characters and normality of Sylow $p$-subgroups

Several refinements of (the normality part of) the celebrated It\^o--Michler theorem were obtained during the last two decades, in which the condition of having $p'$-degree, for a fixed prime $p$, is imposed only on some subsets of complex irreducible characters of a finite group $G$. We prove further extensions of these results, where this condition is now imposed on the irreducible characters which lie above the principal character of a Sylow $p$-subgroup and are either $p$-rational, or strongly real when $p=2$.

math.GR↗
arXiv2024

On the structure of the character degree graphs having diameter three

The structure of the character degree graphs $Δ(G)$, i.e. the prime graphs on the set $\mathrm{cd}(G)$ of the irreducible character degrees of a finite group $G$, such that $G$ is solvable and $Δ(G)$ has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of Mark Lewis.

math.GR↗
arXiv2023

Finite groups with a small proportion of vanishing elements

The function $\mathrm{P}_{\mathbf{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes (as proved in [12]) only values $\frac{m-1}{m}$, for $1 \leq m \leq 6$, in the interval $[0, \mathrm{P}_{\mathbf{v}}(A_7))$.In this paper, we give a complete classification of the finite groups $G$ such that $\mathrm{P}_{\mathbf{v}}(G)=\frac{m-1}{m}$ for $m=1,2,\cdots ,6$.

math.GR↗
arXiv2022

Non-solvable groups whose character degree graph has a cut-vertex. I

Let G be a finite group. Denoting by cd(G) the set of degrees of the irreducible complex characters of G, we consider the character degree graph of G: this is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cd(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in cd(G). In the series of three papers starting with the present one, we analyze the structure of the finite non-solvable groups whose character degree graph possesses a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph.

math.GR↗
arXiv2022

Non-solvable groups whose character degree graph has a cut-vertex. II

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $Δ(G)$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This paper continues the work, started in [7], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. While, in [7], groups with no composition factors isomorphic to ${\rm{PSL}}_2(t^a)$ (for any prime power $t^a\geq 4$) were treated, here we consider the complementary situation in the case when $t$ is odd and $t^a> 5$. The proof of this classification will be then completed in the third and last paper of this series ([8]), that deals with the case $t=2$.

math.GR↗
arXiv2022

On the proportion of vanishing elements in finite groups

We prove that the function $\mathrm{P}_{\mathrm{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes very sparse values in a large segment of the $[0,1]$ interval.

math.GR↗
arXiv2021

On Huppert's Rho-Sigma Conjecture

For an irreducible complex character $χ$ of the finite group $G$, let $π(χ)$ denote the set of prime divisors of the degree $χ(1)$ of $χ$. Denote then by $ρ(G)$ the union of all the sets $π(χ)$ and by $σ(G)$ the largest value of $|π(χ)|$, as $χ$ runs in ${\rm{Irr}}(G)$. The $ρ$-$σ$ conjecture, formulated by Bertram Huppert in the 80's, predicts that $|ρ(G)|\leq 3σ(G)$ always holds, whereas $|ρ(G)|\leq 2σ(G)$ holds if $G$ is solvable; moreover, O. Manz and T.R. Wolf proposed a "strengthened" form of the conjecture in the general case, asking whether $|ρ(G)|\leq 2σ(G)+1$ is true for every finite group $G$. In this paper we study the strengthened $ρ$-$σ$ conjecture for the class of finite groups having a trivial Fitting subgroup: in this context, we prove that the conjecture is true provided $σ(G)\leq 5$, but it is false in general if $σ(G)\geq 6$. Instead, we establish that $|ρ(G)|\leq 3σ(G)-4$ holds for every finite group with a trivial Fitting subgroup and with $σ(G)\geq 6$ (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have $|ρ(G)|\leq 3σ(G)$ whenever $G$ belongs to one particular class including all the finite solvable groups.

math.GR↗
arXiv2019

Groups whose prime graph on class sizes has a cut vertex

Let $G$ be a finite group, and let $Δ(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $Δ(G)$ has a cut vertex.

math.GR↗
arXiv2019

$p$-Power conjugacy classes in $U(n,q)$ and $T(n,q)$

Let $q$ be a $p$-power where $p$ is a fixed prime. In this paper, we look at the $p$-power maps on unitriangular group $U(n,q)$ and triangular group $T(n,q)$. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

math.GR↗
arXiv2018

Bounding the number of vertices in the degree graph of a finite group

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $Δ(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists of the prime divisors of the numbers in ${\rm{cd}}(G)$, two distinct vertices $p$ and $q$ being adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. In this note, we provide an upper bound on the size of ${\rm{V}}(G)$ in terms of the clique number $ω(G)$ (i.e., the maximum size of a subset of ${\rm{V}}(G)$ inducing a complete subgraph) of $Δ(G)$. Namely, we show that $|{\rm{V}}(G)|\leq{\rm{max}}\{2ω(G)+1,\;3ω(G)-4\}$. Examples are given in order to show that the bound is best possible. This completes the analysis carried out in [1] where the solvable case was treated, extends the results in [3,4,9], and answers a question posed by the first author and H.P. Tong-Viet in [4].

math.GR↗
arXiv2018

On the character degree graph of finite groups

Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of [1], where the solvable case was treated.

math.GR↗
arXiv2017

On the character degree graph of solvable groups

Let \(G\) be a finite solvable group, and let \(Δ(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. Pálfy asserts that the complement $\barΔ(G)$ of the graph \(Δ(G)\) does not contain any cycle of length \(3\). In this paper we generalize Pálfy's result, showing that $\barΔ(G)$ does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of \(Δ(G)\) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if \(n\) is the clique number of \(Δ(G)\), then \(Δ(G)\) has at most \(2n\) vertices. This confirms a conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for the famous \emph{\(ρ\)-\(σ\) conjecture} by B. Huppert.

math.GR↗
arXiv2016

Groups whose character degree graph has diameter three

Let \(G\) be a finite group, and let \(Δ(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(Δ(G)\) is connected, the diameter of \(Δ(G)\) is at most \(3\). In the present paper, we provide a description of the finite solvable groups for which the diameter of this graph attains the upper bound. This also enables us to confirm a couple of conjectures proposed by M.L. Lewis.

math.GR↗
arXiv2012

On the maximal number of coprime subdegrees in finite primitive permutation groups

The subdegrees of a transitive permutation group are the orbit lengths of a point stabilizer. For a finite primitive permutation group which is not cyclic of prime order, the largest subdegree shares a non-trivial common factor with each non-trivial subdegree. On the other hand it is possible for non-trivial subdegrees of primitive groups to be coprime, a famous example being the rank 5 action of the small Janko group on 266 points which has subdegrees of lengths 11 and 12. We prove that, for every finite primitive group, the maximal size of a set of pairwise coprime non-trivial subdegrees is at most 2.

math.GR↗
arXiv2011

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H acting completely reducibly on a vector space V: if the orbits containing the vectors a and b have coprime lengths m and n, we prove that the orbit containing a+b has length mn. Such groups H are always reducible if n and m are greater than 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor. In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.

math.GR↗
arXiv2011

A new solvability criterion for finite groups

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

math.GR↗