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Emanuele Pacifici

Publications and source records attributed to Emanuele Pacifici.

At least 19 recordsLinked to original sources
arXiv2026

The Gruenberg-Kegel graph of finite solvable groups that are character-quadratic or semi-rational

A finite group $G$ is said to be semi-rational if the set of generators of each cyclic subgroup of $G$ is contained in at most two $G$-conjugacy classes. This is equivalent to the following condition: for every column of the character table of $G$, the values appearing in the column are contained in a quadratic extension of the field of rational numbers (possibly a different one for each column). When the analogous condition holds for the rows, that is, when the field of values of every irreducible character is contained in a quadratic extension of the rationals, we say that the group is character-quadratic (these groups are often called quadratic rational in the literature). We obtain several results concerning the structure of the Gruenberg-Kegel graph of a finite solvable group that is either character-quadratic or semi-rational. More precisely, we first provide a complete classification of such graphs in the disconnected case. Also, we prove that if the graph has at most three vertices and the group is nontrivial, then it belongs to an explicit list of $20$ graphs (in the semi-rational case, this result is proved under the additional assumption that the order of the group is not divisible by $17$), and all of them are realizable except perhaps one. Finally, we show that if the graph has four vertices, then it must have at least four edges.

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arXiv2026

Groups with a conjugacy class that is the difference of two normal subgroups

We consider finite groups having a conjugacy class that is the difference of two normal subgroups. That is, suppose $G$ is a group and $M$ and $N$ are normal subgroups so that $N < M$, and suppose that there is an element $g \in G$ so that the conjugacy class of $g$ is $M \setminus N$. We find a character-theoretic characterization of this condition, and we determine some structural properties of groups with such a conjugacy class. If we add the condition that $M/N$ is the unique minimal normal subgroup of $G/N$, then we obtain a generalization of a result by S.M. Gagola.

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arXiv2025

The N-prime graph and the Subgroup Isomorphism Problem

We introduce a directed graph related to a group $G$, which we call the N-prime graph $Γ_{\rm{N}}(G)$ of $G$ and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of $Γ_{\rm{N}}(G)$ are the primes $p$ such that $G$ has an element of order $p$, and, for distinct vertices $p$ and $q$, the arc $q\rightarrow p$ is in the graph if and only if $G$ has a subgroup of order $p$ whose normalizer in $G$ has an element of order $q$. Generalizing some known results about the Gruenberg-Kegel graph, we prove that the group $V(\mathbb{Z} G)$ of the units with augmentation $1$ in the integral group ring $\mathbb{Z} G$ has the same N-prime graph as $G$ if $G$ is a finite solvable group, and we reduce to almost simple groups the problem of whether $Γ_{\rm{N}}(V(\mathbb{Z} G))=Γ_{\rm{N}}(G)$ holds for any finite group $G$. We also prove that $Γ_{\rm{N}}(V(\mathbb{Z} G))=Γ_{\rm{N}}(G)$ if $G$ is almost simple with socle either an alternating group, or ${\rm{PSL}}(r^f)$ with $r$ prime and $f\le 2$. Finally, for $G$ solvable we obtain some stronger results which give a contribution to the Subgroup Isomorphism Problem. More precisely, we prove that if $V(\mathbb{Z} G)$ contains a Frobenius subgroup $T$ with kernel of prime order and complement of prime power order, then $G$ contains a subgroup isomorphic to $T$.

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arXiv2025

Groups with conjugacy classes of coprime sizes

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $π$-regular elements for some set of primes $π$, then $x^G y^G$ is a $π$-regular conjugacy class in $G$. The latter statement was previously known for $π$-separable groups $G$ and this generalisation permits us to extend several results concerning the common divisor graph on $p$-regular conjugacy classes, for some prime $p$.

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arXiv2025

Group cosets with all elements of equal order

Let $G$ be a finite group and $N$ a proper, nontrivial, normal subgroup of $G$. If, for every element $x$ of $G$ not lying in $N$, the elements in the coset $xN$ all have the same order as $x$, then we say that $(G,N)$ is an {\it{equal order pair}}. This generalizes the concept of a Camina pair, that was introduced by the first author. In the present paper we study several properties of equal order pairs, showing that in many respects they resemble Camina pairs, but with some important differences.

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arXiv2025

On quadratic rational Frobenius groups

Let $G$ be a finite group and, for a given complex character $χ$ of $G$, let ${\mathbb{Q}}(χ)$ denote the field extension of ${\mathbb{Q}}$ obtained by adjoining all the values $χ(g)$, for $g\in G$. The group $G$ is called quadratic rational if, for every irreducible complex character $χ\in{\rm{Irr}}(G)$, the field ${\mathbb{Q}}(χ)$ is an extension of ${\mathbb{Q}}$ of degree at most $2$. Quadratic rational groups have a nice characterization in terms of the structure of the group of central units in their integral group ring, and in fact they generalize the well-known concept of a cut group (i.e., a finite group whose integral group ring has a finite group of central units). In this paper we classify the Frobenius groups that are quadratic rational, a crucial step in the project of describing the Gruenberg-Kegel graphs associated to quadratic rational groups. It turns out that every Frobenius quadratic rational group is uniformly semi-rational, i.e., it satisfies the following property: all the generators of any cyclic subgroup of $G$ lie in at most two conjugacy classes of $G$, and these classes are permuted by the same element of the Galois group ${\rm{Gal}}({\mathbb{Q}}_{|G|}/{\mathbb{Q}})$ (in general, every cut group is uniformly semi-rational, and every uniformly semi-rational group is quadratic rational). We will also see that the class of groups here considered coincides with the one studied in [4], thus the main result of this paper also completes the analysis carried out in [4].

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arXiv2024

On non self-normalizing subgroups

Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_n$ and its interplay with solvability and nilpotency. We first show that if $G$ belongs to $D_n$ with $n \le 3$, then $G$ is solvable of derived length at most 2. We also show that $A_5$ is the unique nonsolvable group in $D_4$, and that $SL_2(3)$ is the unique solvable group in $D_4$ whose derived length is larger than 2. For a group $G$, we define $D(G)$ to be the number of conjugacy classes of nontrivial subgroups that are not self-normalizing. We determine the relationship between $D(H \times K)$ and $D(H)$ and $D(K)$. We show that if $G$ is nilpotent and lies in $D_n$, then $G$ has nilpotency class at most $n/2$ and its derived length is at most $\log_2 (n/2) + 1$. We consider $D_n$ for several classes of Frobenius groups, and we use this classification to classify the groups in $D_0$, $D_1$, $D_2$, and $D_3$. Finally, we show that if $G$ is solvable and lies in $D_n$ with $n \ge 3$, then $G$ has derived length at most the minimum of $n-1$ and $3 \log_2 (n+1) + 9$.

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arXiv2024

On common zeros of characters of finite groups

Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $χ\in\text{Irr}(G)$ such that $χ(g)=0$ (i.e., $g$ is a zero of $χ$) and, in this case, the conjugacy class $g^G$ of $g$ in $G$ is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group $G$ such that every non-linear $χ\in\text{Irr}(G)$ vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.

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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.

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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↗
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.

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arXiv2020

On solvable groups with one vanishing class size

Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylow $2$-subgroup} (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size \emph{which is either a prime power or square-free}.

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↗
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].

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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 vanishing class sizes in finite groups

Let $G$ be a finite group. An element $g$ of $G$ is called a vanishing element if there exists an irreducible character $χ$ of $G$ such that $χ(g) = 0$; in this case, we say that the conjugacy class of $g$ is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group.

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.

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