Víctor Sotomayor

The first author is supported by Proyecto Prometeo II/2015/011, and the third author acknowledges the doctorate scholarship “Ayudas para la contratacion de personal investigador en formacion de caracter predoctoral granted … The first author is supported by Proyecto Prometeo II/2015/011, and the third author acknowledges the doctorate scholarship “Ayudas para la contratacion de personal investigador en formacion de caracter predoctoral granted by Generalitat Valenciana, Spain.

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On zeros of irreducible characters lying in a normal subgroup

2020-01-29

María José Felipe , Nicola Grittini , Víctor Sotomayor

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Groups whose prime graph on class sizes has a cut vertex

2021-09-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Structural Criteria in Factorised Groups Via Conjugacy Class Sizes

2019-04-11

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

We report on current activity regarding structural properties of finite factorised groups provided that the conjugacy class sizes of some elements in the factors have certain arithmetical conditions. We report on current activity regarding structural properties of finite factorised groups provided that the conjugacy class sizes of some elements in the factors have certain arithmetical conditions.

On products of groups and indices not divisible by a given prime

2020-07-22

María José Felipe , Л. С. Казарин , A. Martı́nez-Pastor +1 more

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Finite groups whose prime graph on class sizes is a block square

2022-04-01

Víctor Sotomayor

Let G be a finite group, and let Δ(G) be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are … Let G be a finite group, and let Δ(G) be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of G, and two distinct vertices p, q are adjacent if and only if pq divides some class size of G. In this paper, we characterize the structure of those groups G whose prime graph Δ(G) is a block square.

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

2022-01-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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 … 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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Products of groups and class sizes of $π$-elements

2018-01-01

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

We provide structural criteria for some finite factorised groups $G = AB$ when the conjugacy class sizes in $G$ of certain $\pi$-elements in $A\cup B$ are either $\pi$-numbers or $\pi'$-numbers, … We provide structural criteria for some finite factorised groups $G = AB$ when the conjugacy class sizes in $G$ of certain $\pi$-elements in $A\cup B$ are either $\pi$-numbers or $\pi'$-numbers, for a set of primes $\pi$. In particular, we extend for products of groups some earlier results.

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On zeros of irreducible characters lying in a normal subgroup

2019-01-01

María José Felipe , Nicola Grittini , Víctor Sotomayor

Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $\chi(g)\neq 0$ for all irreducible characters $\chi$ … Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $\chi(g)\neq 0$ for all irreducible characters $\chi$ 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 also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.

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Conjugacy classes and factorised groups

2019-07-05

Víctor Sotomayor

la mia permanenza lì un dono. la mia permanenza lì un dono.

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Groups whose prime graph on class sizes has a cut vertex

2019-12-20

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose … Let $G$ be a finite group, and let $\Delta(G)$ 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 $\Delta(G)$ has a cut vertex.

On G-character tables for normal subgroups

2022-03-30

María José Felipe , M. D. Pérez‐Ramos , Víctor Sotomayor

Let N be a normal subgroup of a finite group G. From a result due to Brauer, it can be derived that the character table of G contains square submatrices … Let N be a normal subgroup of a finite group G. From a result due to Brauer, it can be derived that the character table of G contains square submatrices which are induced by the G-conjugacy classes of elements in N and the G-orbits of irreducible characters of N. In the present paper, we provide an alternative approach to this fact through the structure of the group algebra. We also show that such matrices are non-singular and become a useful tool to obtain information of N from the character table of G.

Finite groups whose prime graph on class sizes is a block square

2021-01-01

Víctor Sotomayor

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are … Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, and two distinct vertices $p, q$ are adjacent if and only if $pq$ divides some class size of $G$. In this paper, we characterise the structure of those groups $G$ whose prime graph $\Delta(G)$ is a block square.

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On products of groups and indices not divisible by a given prime

2020-01-01

María-José Felipe , Л. С. Казарин , A. Martı́nez-Pastor +1 more

Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. We prove that $p$ does not divide the conjugacy class … Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. We prove that $p$ does not divide the conjugacy class size (index) of each $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is $p$-decomposable, i.e. $G=O_p(G) \times O_{p'}(G)$.

Groups whose prime graph on class sizes has a cut vertex

2019-01-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Finite groups and the class-size prime graph revisited

2022-01-01

Víctor Sotomayor

We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph. We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph.

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Non-solvable Groups whose Character Degree Graph has a Cut-Vertex. I

2023-06-21

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

Abstract Let G be a finite group. Denoting by $$\textrm{cd}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>cd</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> the set of degrees of the irreducible complex characters of G … Abstract Let G be a finite group. Denoting by $$\textrm{cd}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>cd</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\textrm{cd}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>cd</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and two distinct vertices p , q are adjacent if and only if pq divides some number in $$\textrm{cd}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>cd</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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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Determining hypercentral Hall subgroups in finite groups

2024-03-01

Víctor Sotomayor

Let $G$ be a finite group, and let $\pi$ be a set of primes. The aim of this paper is to obtain some results concerning how much information about the … Let $G$ be a finite group, and let $\pi$ be a set of primes. The aim of this paper is to obtain some results concerning how much information about the $\pi$-structure of $G$ can be gathered from the knowledge of the lengths of conjugacy classes of its $\pi$-elements and of their multiplicities. Among other results, we prove that this multiset of class lengths determine whether $G$ has a hypercentral Hall $\pi$-subgroup.

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Groups whose common divisor graph on $p$-regular classes has diameter three

2024-07-13

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $\Gamma_p(G)$ be the common divisor graph built on the set of non-central conjugacy classes of $p$-regular elements … Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $\Gamma_p(G)$ be the common divisor graph built on the set of non-central conjugacy classes of $p$-regular elements of $G$: this is the graph whose vertices are the conjugacy classes of those non-central elements of $G$ such that $p$ does not divide their orders, and two distinct vertices are adjacent if and only if the greatest common divisor of their lengths is strictly greater than one. The aim of this paper is twofold: to positively answer an open question concerning the maximum possible distance in $\Gamma_p(G)$ between a vertex with maximal cardinality and any other vertex, and to study the $p$-structure of $G$ when $\Gamma_p(G)$ has diameter three.

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Determining Hypercentral Hall Subgroups in Finite Groups

2024-08-12

Víctor Sotomayor

Abstract Let G be a finite group, and let $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> be a set of primes. The aim of this paper is to obtain some results … Abstract Let G be a finite group, and let $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> be a set of primes. The aim of this paper is to obtain some results concerning how much information about the $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> -structure of G can be gathered from the knowledge of the sizes of conjugacy classes of its $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> -elements and of their multiplicities. Among other results, we prove that this multiset of class sizes determines whether G has a hypercentral Hall $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> -subgroup.

On $G$-character tables for normal subgroups

2024-09-17

María José Felipe , M. D. Pérez‐Ramos , Víctor Sotomayor

Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices … Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices which are induced by the $G$-conjugacy classes of elements in $N$ and the $G$-orbits of irreducible characters of $N$. In the present paper, we provide an alternative approach to this fact through the structure of the group algebra. We also show that such matrices are non-singular and become a useful tool to obtain information of $N$ from the character table of $G$.

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Classification of groups with triangle-free graphs on p-regular classes

2024-08-05

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

Let $p$ be a prime. In this paper we classify the $p$-structure of those finite $p$-separable groups such that, given any three non-central conjugacy classes of $p$-regular elements, two of … Let $p$ be a prime. In this paper we classify the $p$-structure of those finite $p$-separable groups such that, given any three non-central conjugacy classes of $p$-regular elements, two of them necessarily have coprime lengths.

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A completeness criterion for the common divisor graph on $p$-regular class sizes

2024-12-12

Víctor Sotomayor

Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $\Gamma_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of … Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $\Gamma_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of those non-central elements of $G$ such that $p$ does not divide their order, and two distinct vertices are adjacent if and only if they are not coprime. In this note we prove that if $\Gamma_p(G)$ is a $k$-regular graph with $k\geq 1$, then it is a complete graph with $k+1$ vertices. We also pose a conjecture regarding the order of products of $p$-regular elements with coprime conjugacy class sizes, whose validity would enable to drop the $p$-separability hypothesis.

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Codes in algebras of direct products of groups and associated quantum codes

2024-12-12

Matheus Rolim Sales , Xaro Soler–Escrivà , Víctor Sotomayor

In this paper we obtain the Wedderburn-Artin decomposition of a semisimple group algebra associated to a direct product of finite groups. We also provide formulae for the number of all … In this paper we obtain the Wedderburn-Artin decomposition of a semisimple group algebra associated to a direct product of finite groups. We also provide formulae for the number of all possible group codes, and their dimensions, that can be constructed in a group algebra. As particular cases, we present the complete algebraic description of the group algebra of any direct product of groups whose direct factors are cyclic, dihedral, or generalised quaternion groups. Finally, in the specific case of semisimple dihedral group algebras, we give a method to build quantum error-correcting codes, based on the CSS construction.

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Groups Whose Common Divisor Graph on p-Regular Classes has Diameter Three

2024-12-23

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

Abstract Let G be a finite p -separable group, for some fixed prime p . Let $$\Gamma _p(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> … Abstract Let G be a finite p -separable group, for some fixed prime p . Let $$\Gamma _p(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the common divisor graph built on the set of non-central conjugacy classes of p -regular elements of G : this is the graph whose vertices are the conjugacy classes of those non-central elements of G such that p does not divide their orders, and two distinct vertices are adjacent if and only if the greatest common divisor of their lengths is strictly greater than one. The aim of this paper is twofold: to positively answer an open question concerning the maximum possible distance in $$\Gamma _p(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> between a vertex with maximal cardinality and any other vertex, and to study the p -structure of G when $$\Gamma _p(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has diameter three.

Groups with triangle‐free graphs on p$p$‐regular classes

2025-05-04

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

Abstract Let be a prime. In this paper, we classify the ‐structure of those finite ‐separable groups such that, given any three non‐central conjugacy classes of ‐regular elements, two of … Abstract Let be a prime. In this paper, we classify the ‐structure of those finite ‐separable groups such that, given any three non‐central conjugacy classes of ‐regular elements, two of them necessarily have coprime lengths.

Groups with triangle‐free graphs on p$p$‐regular classes

2025-05-04

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

Groups Whose Common Divisor Graph on p-Regular Classes has Diameter Three

2024-12-23

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

A completeness criterion for the common divisor graph on $p$-regular class sizes

2024-12-12

Víctor Sotomayor

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Codes in algebras of direct products of groups and associated quantum codes

2024-12-12

Matheus Rolim Sales , Xaro Soler–Escrivà , Víctor Sotomayor

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On $G$-character tables for normal subgroups

2024-09-17

María José Felipe , M. D. Pérez‐Ramos , Víctor Sotomayor

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Determining Hypercentral Hall Subgroups in Finite Groups

2024-08-12

Víctor Sotomayor

Classification of groups with triangle-free graphs on p-regular classes

2024-08-05

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

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Groups whose common divisor graph on $p$-regular classes has diameter three

2024-07-13

María José Felipe , M. K. Jean‐Philippe , Víctor Sotomayor

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Determining hypercentral Hall subgroups in finite groups

2024-03-01

Víctor Sotomayor

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Non-solvable Groups whose Character Degree Graph has a Cut-Vertex. I

2023-06-21

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Finite groups whose prime graph on class sizes is a block square

2022-04-01

Víctor Sotomayor

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On G-character tables for normal subgroups

2022-03-30

María José Felipe , M. D. Pérez‐Ramos , Víctor Sotomayor

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

2022-01-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Finite groups and the class-size prime graph revisited

2022-01-01

Víctor Sotomayor

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Groups whose prime graph on class sizes has a cut vertex

2021-09-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Finite groups whose prime graph on class sizes is a block square

2021-01-01

Víctor Sotomayor

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On products of groups and indices not divisible by a given prime

2020-07-22

María José Felipe , Л. С. Казарин , A. Martı́nez-Pastor +1 more

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On zeros of irreducible characters lying in a normal subgroup

2020-01-29

María José Felipe , Nicola Grittini , Víctor Sotomayor

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On products of groups and indices not divisible by a given prime

2020-01-01

María-José Felipe , Л. С. Казарин , A. Martı́nez-Pastor +1 more

Groups whose prime graph on class sizes has a cut vertex

2019-12-20

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

Products of Groups and Class Sizes of $$\pi $$-Elements

2019-12-12

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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The prime graph on class sizes of a finite group has a bipartite complement

2019-10-08

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Conjugacy classes and factorised groups

2019-07-05

Víctor Sotomayor

la mia permanenza lì un dono. la mia permanenza lì un dono.

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Structural Criteria in Factorised Groups Via Conjugacy Class Sizes

2019-04-11

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

On zeros of irreducible characters lying in a normal subgroup

2019-01-01

María José Felipe , Nicola Grittini , Víctor Sotomayor

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Groups whose prime graph on class sizes has a cut vertex

2019-01-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Zeros of irreducible characters in factorised groups

2018-06-21

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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On finite groups with square-free conjugacy class sizes

2018-06-01

María-José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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Products of groups and class sizes of $π$-elements

2018-01-01

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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Prime Power Indices in Factorised Groups

2017-10-19

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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Square-free class sizes in products of groups

2017-08-18

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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Coauthor	Papers Together
María José Felipe	15
A. Martı́nez-Pastor	9
Emanuele Pacifici	6
Silvio Dolfi	6
Lucía Sanus	6
M. K. Jean‐Philippe	4
Nicola Grittini	2
María-José Felipe	2
Л. С. Казарин	2
M. D. Pérez‐Ramos	2
Matheus Rolim Sales	1
Xaro Soler–Escrivà	1

Mutually permutable products and conjugacy classes

2012-04-20

A. Ballester‐Bolinches , John Cossey , Yangming Li

On the orders of zeros of irreducible characters

2008-11-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus +1 more

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Indices of elements and normal structure of finite groups

2004-12-13

Yakov Berkovich , Л. С. Казарин

Arithmetical Conditions on the Length of the Conjugacy Classes of a Finite Group

1995-06-01

Silvio Dolfi

Notes on the length of conjugacy classes of finite groups

2004-10-12

Xiaolei Liu , Yanming Wang , Huaquan Wei

Implications of conjugacy class size

1998-01-01

Rachel D. Camina , A. R. Camina

Finite Soluble Groups

1992-12-31

Klaus Doerk , Trevor Hawkes

Character Theory of Finite Groups

1976-01-01

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Prime powers as conjugacy class lengths of π-elements

2004-04-01

Antonio Beltrán , María José Felipe

Let G be a finite group and π an arbitrary set of primes. We investigate the structure of G when the lengths of the conjugacy classes of its π-elements are … Let G be a finite group and π an arbitrary set of primes. We investigate the structure of G when the lengths of the conjugacy classes of its π-elements are prime powers. Under this condition, we show that such lengths are either powers of just one prime or exactly {1, q a , r b }, with q and r two distinct primes lying in π and a , b > 0. In the first case, we obtain certain properties of the normal structure of G , and in the second one, we provide a characterisation of the structure of G .

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Nilpotent and abelian Hall subgroups in finite groups

2015-07-10

Antonio Beltrán , María José Felipe , Gunter Malle +5 more

We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups that can easily be verified using the character table. We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups that can easily be verified using the character table.

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Non-vanishing elements in finite groups

2016-06-01

Julian Brough

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On the length of the conjugacy classes of finite groups

1990-05-01

David Chillag , Marcel Herzog

Remarks on the length of conjugacy classes of finite groups

1999-01-01

John Cossey , Yanming Wang

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by … Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p 2 for some prime p. We generalise and provide simplified proofs for some earlier results.

Finite Group Elements where No Irreducible Character Vanishes

1999-12-01

I. M. Isaacs , Gabriel Navarro , Thomas R. Wolf

Prime Power Indices in Factorised Groups

2017-10-19

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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THE INFLUENCE OF CONJUGACY CLASS SIZES ON THE STRUCTURE OF FINITE GROUPS: A SURVEY

2011-12-01

A. R. Camina , Rachel D. Camina

The importance of conjugacy classes for the structure of finite groups was recognised very early in the study of groups. In this survey we consider the results from the many … The importance of conjugacy classes for the structure of finite groups was recognised very early in the study of groups. In this survey we consider the results from the many articles which have developed this topic and examined the influence of conjugacy class sizes or the number of conjugacy classes on the structure of finite groups. Whilst we begin by mentioning the early results of Sylow and Burnside, our major objective is to highlight the more recent work and present some interesting questions which we hope will inspire further research.

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Square-free class sizes in products of groups

2017-08-18

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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Incomplete vertices in the prime graph on conjugacy class sizes of finite groups

2012-11-30

Carlo Casolo , Silvio Dolfi , Emanuele Pacifici +1 more

ON VANISHING CRITERIA THAT CONTROL FINITE GROUP STRUCTURE II

2018-08-01

Julian Brough , Qingjun Kong

The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra 458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it … The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra 458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.

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On vanishing criteria that control finite group structure

2016-04-11

Julian Brough

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Solvable groups whose prime divisor character degree graphs are 1-connected

2019-03-01

Mark L. Lewis , Qingyun Meng

Groups whose vanishing class sizes are not divisible by a given prime

2010-03-23

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus

On finite groups with square-free conjugacy class sizes

2018-06-01

María-José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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On the structure of a mutually permutable product of finite groups

2018-02-22

J Cossey , Youyou LI

Groups whose prime graph on conjugacy class sizes has few complete vertices

2012-05-07

Carlo Casolo , Silvio Dolfi , Emanuele Pacifici +1 more

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Zeros of characters of finite groups

2000-01-16

Gunter Malle , Gabriel Navarro , Jørn B. Olsson

There are some variations of theorems A and B which are simply not true. For instance, if χ in Irr(G) has degree divisible by p, then there does not necessarily … There are some variations of theorems A and B which are simply not true. For instance, if χ in Irr(G) has degree divisible by p, then there does not necessarily exist a p-element on which χ vanishes. It is enough to consider L2(11) with any character of degree 10 and p = 2. It is also not true that if χ vanishes on some element x, then χ has to vanish on some p-part of x. For instance, if G is M11, then χ has an irreducible character of degree 11, vanishing on an element of order 6 and which is nonzero on 2and 3-elements. Thirdly, it is not true that a nonlinear character has to vanish on some element of prime order, as shown by any quaternion group. Interestingly enough, this seems to be the case for simple groups (and we do prove this for the groups of Lie type and the sporadic groups).

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On Finite Groups with Given Conjugate Types I

1953-10-01

Noboru Ιτο

Let G be a finite group. Let n 1 , n 2 ,…, n r , where n 1 >n 2 > … > n r = 1, be all … Let G be a finite group. Let n 1 , n 2 ,…, n r , where n 1 >n 2 > … > n r = 1, be all the numbers each of which is the index of the centralizer of some element of G in G. We call the vector ( n 1 , n 2 ,…, n r ) the conjugate type vector of G. A group with the conjugate type vector ( n 1 , n 2 ,…, n r ) is said to be a group of type ( n 1 , n 2 ,…, n r ).

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Products of primes in conjugacy class sizes and irreducible character degrees

2009-11-01

Carlo Casolo , Silvio Dolfi

On independent sets in the class graph of a finite group

2005-11-07

Silvio Dolfi

Products of Groups and Class Sizes of $$\pi $$-Elements

2019-12-12

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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Zeros of irreducible characters in factorised groups

2018-06-21

María José Felipe , A. Martı́nez-Pastor , Víctor Sotomayor

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Character theory of finite groups

1999-07-01

I. M. Isaacs

Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur … Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.

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ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

2013-06-17

Guohua Qian , Yanming Wang

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, … Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, |G|) = 1 and p 2 does not divide |x G | for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/O p (G) is elementary abelian. (2) Suppose that G is p-solvable. If p p-1 does not divide |x G | for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If p p-1 does not divide χ(1) for any χ ∈ Irr (G), then both the p-length and p′-length of G are at most 2.

Group elements of prime power index

1953-01-01

Reinhold Baer

The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to -g in G. The significance of elements of prime power index … The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to -g in G. The significance of elements of prime power index is best recognized once one remembers Wielandt's Theorem that elements whose order and index are powers of the same prime p are contained in a normal subgroup of order a power of p and Burnside's Theorem asserting the absence of elements of prime power index, not 1, in simple groups. From Burnside's Theorem one deduces easily that a group without proper characteristic subgroups contains an element, not 1, whose index is a power of a prime if and only if this group is abelian. In this result it suffices to assume the absence of proper fully invariant subgroups, since we can prove [in ?2] the rather surprising result that a [finite] group does not possess proper fully invariant subgroups if and only if it does not possess proper characteristic subgroups. A deeper insight will be gained if we consider groups which contain many elements of prime power index. We show [in ?5 ] that the elements of order a power of p form a direct factor if, and only if, their indices are powers of p too; and nilpotency is naturally equivalent to the requirement that this property holds for every prime p. More difficult is the determination of groups with the property that every element of prime power order has also prime power index [?3]. It follows from Burnside's Theorem that such groups are soluble; and it is clear that a group has this property if it is the direct product of groups of relatively prime orders which are either p-groups or else have orders divisible by only two different primes and furthermore have abelian Sylow subgroups. But we are able to show conversely that every group with the property under consideration may be represented in the fashion indicated. In ?5 we study the so-called hypercenter. This characteristic subgroup has been defined in various ways: as the terminal member of the ascending central chain or as the smallest normal subgroup modulo which the center is 1. We may add here such further characterizations as the intersection of all the normalizers of all the Sylow subgroups or as the intersection of all the maximal nilpotent subgroups; and the connection with the index problem is obtained by showing that a normal subgroup is part of the hypercenter if, and only if, its elements of order a power of p have also index a powrer of p. Notation. All the groups under consideration will be finite. An element [group] is termed primary, if its order is a prime power;

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On vanishing class sizes in finite groups

2017-07-24

Mariagrazia Bianchi , Julian Brough , Rachel D. Camina +1 more

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On Zeros of Characters of Finite Groups

2018-01-01

Silvio Dolfi , Emanuele Pacifici , Lucía Sanus

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Character degree graphs that are complete graphs

2006-08-31

Mariagrazia Bianchi , David Chillag , Mark L. Lewis +1 more

Let $G$ be a finite group, and write $\operatorname {cd}(G)$ for the set of degrees of irreducible characters of $G$. We define $\Gamma (G)$ to be the graph whose vertex … Let $G$ be a finite group, and write $\operatorname {cd}(G)$ for the set of degrees of irreducible characters of $G$. We define $\Gamma (G)$ to be the graph whose vertex set is $\operatorname {cd}(G)-\{1\}$, and there is an edge between $a$ and $b$ if $(a,b)>1$. We prove that if $\Gamma (G)$ is a complete graph, then $G$ is a solvable group.

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Relative Brauer groups II.

1981-11-01

Murray Schacher , Burton Fein , William M. Kantor

Two remarks about non-vanishing elements in finite groups

2016-06-01

Matthias Grüninger

On the character degree graph of finite groups

2019-02-18

Zeinab Akhlaghi , Carlo Casolo , Silvio Dolfi +2 more

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Finite Groups with a Disconnectedp-Regular Conjugacy Class Graph

2004-10-01

Antonio Beltrán , María José Felipe

Abstract Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ p (G) whose vertices are the non-central p-regular conjugacy classes of … Abstract Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ p (G) whose vertices are the non-central p-regular conjugacy classes of G and connect two vertices by an edge if their cardinalities have a common prime divisor. In this note we study the structure and arithmetical properties of the p-regular class sizes in p-solvable groups G having Γ p (G) disconnected.

ON THE DIAMETER OF A p-REGULAR CONJUGACY CLASS GRAPH OF FINITE GROUPS

2002-12-11

Antonio Beltrán , María José Felipe

ABSTRACT Let be a finite -solvable group. Attach to the following graph : its vertices are the non-central conjugacy classes of -regular elements of , and two vertices are connected … ABSTRACT Let be a finite -solvable group. Attach to the following graph : its vertices are the non-central conjugacy classes of -regular elements of , and two vertices are connected by an edge if their cardinalities are not coprime. We prove that the number of connected components of is at most 2. When is connected, then the diameter of the graph is at most 3, and when is disconnected, then each of the two components is a complete graph.

On a Graph Related to Conjugacy Classes of Groups

1990-11-01

Edward A. Bertram , Marcel Herzog , Avinoam Mann

Let G be a finite group. Attach to G the following graph Γ: its vertices are the non-central conjugacy classes of G, and two vertices are connected if their cardinalities … Let G be a finite group. Attach to G the following graph Γ: its vertices are the non-central conjugacy classes of G, and two vertices are connected if their cardinalities are not coprime. Denote by n(Γ) the number of the connecte components of Γ. We prove that n(Γ) ⩽ 2 for all finite groups, and we completely characterize groups with n(Γ) = 2. When Γ is connected, then the diameter of the graph is at most 4. For simple non-abelian finite groups, the graph is complete. Similar results are proved for infinite FC-groups.

Characterizing normal Sylow p-subgroups by character degrees

2012-08-29

Gunter Malle , Gabriel Navarro

Abelian Sylow subgroups in a finite group, II

2014-09-12

Gabriel Navarro , Ronald Solomon , Pham Huu Tiep

On the supersolvability of finite groups

1989-10-01

M. Asaad , Ayesha Shaalan

Character Theory of Finite Groups

1998-12-31

Bertram Huppert

Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small … Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small groups products of representation and characters on the number of solutions gm =1 in a group a theorem of A. Hurwitz on multiplicative sums of squares permutation representations and characters the class number real characters and real representations Coprime action groups pa qb Fronebius groups induced characters Brauer's permutation lemma and Glauberman's character correspondence Clifford theory 1 projective representations Clifford theory 2 extension of characters Degree pattern and group structure monomial groups representation of wreath products characters of p-groups groups with a small number of character degrees linear groups the degree graph groups all of whose character degrees are primes two special degree problems lengths of conjugacy classes R. Brauer's theorem on the character ring applications of Brauer's theorems Artin's induction theorem splitting fields the Schur index integral representations three arithmetical applications small kernels and faithful irreducible characters TI-sets involutions groups whose Sylow-2-subgroups are generalized quaternion groups perfect Fronebius complements. (Part contents).

On conditional permutability and factorized groups

2013-01-07

M. Arroyo-Jordá , P. Arroyo-Jordá , A. Martı́nez-Pastor +1 more

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Group classes and mutually permutable products

2005-08-30

James C. Beidleman , Hermann Heineken

On elements of prime-power index in finite groups

2009-09-20

A. R. Camina , Pavel Shumyatsky , Carmela Sica