Let G be a finite group. We show that when the conjugacy class sizes of G are {1, m, n, mn}, with m and n positive integers such that (m, …
Let G be a finite group. We show that when the conjugacy class sizes of G are {1, m, n, mn}, with m and n positive integers such that (m, n) = 1, then G is solvable. As a consequence, we obtain that G is nilpotent and that m = pa and n = qb for two primes p and q.
Abstract Let G be a finite group. We prove that if the set of p -regular conjugacy class sizes of G has exactly two elements, then G has Abelian p …
Abstract Let G be a finite group. We prove that if the set of p -regular conjugacy class sizes of G has exactly two elements, then G has Abelian p -complement or G = PQ × A , with P ∈ Syl p ( G ), Q ∈ Syl q ( G ) and A Abelian.
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.
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 .
Let G be a finite p -solvable group for a fixed prime p . We determine the structure of G when the set of p -regular conjugacy class sizes of …
Let G be a finite p -solvable group for a fixed prime p . We determine the structure of G when the set of p -regular conjugacy class sizes of G is {1, m } for an arbitrary integer m > 1.
Abstract Let G be a finite p -solvable group for a fixed prime p . We study how certain arithmetical conditions on the set of p -regular conjugacy class sizes …
Abstract Let G be a finite p -solvable group for a fixed prime p . We study how certain arithmetical conditions on the set of p -regular conjugacy class sizes of G influence the p -structure of G . In particular, the structure of the p -complements of G is described when this set is {1, m, n } for arbitrary coprime integers m, n > 1. The structure of G is determined when the noncentral p -regular class lengths are consecutive numbers and when all of them are prime powers.
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.
We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and …
We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and that the set of class sizes is exactly {1, m, n, mk}, where m, n > 1 are coprime numbers and k > 1 is a divisor of n.
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.
If $G$ is a finite group and $N$ is a normal subgroup of $G$ with two $G$-conjugacy class sizes of elements of prime power order, then we show that $N$ …
If $G$ is a finite group and $N$ is a normal subgroup of $G$ with two $G$-conjugacy class sizes of elements of prime power order, then we show that $N$ is nilpotent.
Let G be a finite group and suppose that the set of conjugacy class sizes of G is {1, m, mn}, where m, n > 1 are coprime. We prove …
Let G be a finite group and suppose that the set of conjugacy class sizes of G is {1, m, mn}, where m, n > 1 are coprime. We prove that m = p for some prime p dividing n – 1. We also show that G has an abelian normal p-complement and that if P is a Sylow p-subgroup of G, then |P′| = p and |P/Z(G)p| = p2. We obtain other properties and determine completely the structure of G.
Abstract Let G be a finite p -solvable group. We prove that if G has exactly two conjugacy class sizes of p′ -elements of prime power order, say 1 and …
Abstract Let G be a finite p -solvable group. We prove that if G has exactly two conjugacy class sizes of p′ -elements of prime power order, say 1 and m , then m = p a q b , for two distinct primes p and q , and G either has an abelian p -complement or G = PQ × A , with P and Q a Sylow p -subgroup and a Sylow q -subgroup of G , respectively, and A is abelian. In particular, we provide a new extension of Itô’s theorem on groups having exactly two class sizes for elements of prime power order.
It is shown that if the set of conjugacy class sizes of a finite group G is {1, m, n, mn}, where m, n are positive integers which do not …
It is shown that if the set of conjugacy class sizes of a finite group G is {1, m, n, mn}, where m, n are positive integers which do not divide each other, then G is up to central factors a {p, q}-group. In particular, G is solvable.
Abstract In a coprime action, we study character correspondences between the invariant characters of the group and those of subgroups containing the fixed points subgroup. Some character degree consequences are …
Abstract In a coprime action, we study character correspondences between the invariant characters of the group and those of subgroups containing the fixed points subgroup. Some character degree consequences are obtained.
Abstract Let Gbe a finite p-solvable group. Let us consider the graph Γ* p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of …
Abstract Let Gbe a finite p-solvable group. Let us consider the graph Γ* p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of p-regular elements of G and two primes are joined by an edge if there exists such a class whose size is divisible by both primes. Suppose that Γ p *(G) is a connected graph, then we prove that the diameter of this graph is at most 3 and this is the best bound.
Abstract Let N be a normal subgroup of a group G and let p be a prime. We prove that if the p -part of | x | G is …
Abstract Let N be a normal subgroup of a group G and let p be a prime. We prove that if the p -part of | x | G is a constant for every prime-power order element x ∈ N ∖Z( N ), then N is solvable and has normal p -complement.
We shall assume that any group is finite. One of the classic problems in Group Theory is to study how the structure of a group G determines properties on its …
We shall assume that any group is finite. One of the classic problems in Group Theory is to study how the structure of a group G determines properties on its conjugacy class sizes and reciprocally how these class sizes influence the structure of G. During the nineties several authors studied this relation by defining and studying two graphs associated to the conjugacy class sizes. In 1990 (see [8]), E. Bertram, M. Herzog and A. Mann defined a graph Γ (G) as follows: the vertices of Γ (G) are represented by the non-central conjugacy classes of G and two vertices C and D are connected by an edge if |C| and |D| have a common prime divisor. Later, this graph was studied in [12] and also used in [9] to obtain properties on the structure of G when some arithmetical conditions are imposed on the conjugacy class sizes. On the other hand, in 1995, S. Dolfi [14] studied a dual graph, Γ*(G), defined in the following way: the set of vertices are the primes dividing some conjugacy class size of G and two primes r and s are joined by an edge if rs divides some conjugacy class size of G. Independently, G. Alfandary also obtained some properties of these graphs (see [1]).
Abstract Let G be a finite group and let be the conjugacy class of an element x of G . In this paper, it is proved that if N is …
Abstract Let G be a finite group and let be the conjugacy class of an element x of G . In this paper, it is proved that if N is a normal subgroup of G such that the coset is the union of K and (the conjugacy class of the inverse of x ), then N and the subgroup are solvable. As an application, we prove that if there exists a natural number such that , then is solvable.
Let G be a finite solvable group. We prove that any prime dividing any irreducible π-partial character degree of G divides the size of some conjugacy class of π-elements of …
Let G be a finite solvable group. We prove that any prime dividing any irreducible π-partial character degree of G divides the size of some conjugacy class of π-elements of G. Under certain hypothesis, we show that if two distinct primes r and s both divide some irreducible π-partial character degree, then there exists a conjugacy class of π-elements whose size is divisible by rs.
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ . We determine the structure of $N$ when the diameter of the graph associated to …
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ . We determine the structure of $N$ when the diameter of the graph associated to the $G$ -conjugacy classes contained in $N$ is as large as possible, that is, equal to three.
Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see …
Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see text]. We show that, for any positive integers [Formula: see text] and [Formula: see text], there exist finitely many finite groups [Formula: see text], up to isomorphism, having a normal subgroup [Formula: see text] of index [Formula: see text] which contains exactly [Formula: see text] non-central [Formula: see text]-conjugacy classes. Upper bounds for the orders of [Formula: see text] and [Formula: see text] are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few [Formula: see text]-classes. We also study the related problems when we consider only the set of [Formula: see text]-classes of prime-power order elements contained in a normal subgroup.
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. …
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.
Products of conjugacy classes is a well-established theme in Group Theory with open conjectures. We summarize known and new results concerning the influence that the product of two conjugacy classes …
Products of conjugacy classes is a well-established theme in Group Theory with open conjectures. We summarize known and new results concerning the influence that the product of two conjugacy classes exerts on the structure of a finite group. We add several open questions in order to inspire the reader to solve them and develop new techniques of research.
We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes.We show that if the set of G-class sizes of N …
We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes.We show that if the set of G-class sizes of N is {1, m, mp a }, with p a prime not dividing m, then N is solvable.Thus, we get a partial extension for normal subgroups on N. Itô's theorem on the solvability of groups having exactly three class sizes.
Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellon, who is financially supported by a predoctoral grant …
Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellon, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B-2015-77.
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.
The structure of finite groups has a significant influence on the conjugacy class sizes and reciprocally is also influenced by them. In this paper we present some classic and recent …
The structure of finite groups has a significant influence on the conjugacy class sizes and reciprocally is also influenced by them. In this paper we present some classic and recent contributions which have been obtained during the last forty years related to the structure and properties of those groups having few conjugacy class sizes.
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.
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.
A theorem of Z. Arad and E. Fisman establishes that if $A$ and $B$ are two conjugacy classes of a finite group $G$ such that either $AB=A\cup B$ or $AB=A^{-1} …
A theorem of Z. Arad and E. Fisman establishes that if $A$ and $B$ are two conjugacy classes of a finite group $G$ such that either $AB=A\cup B$ or $AB=A^{-1} \cup B$, then $G$ cannot be non-abelian simple. We demonstrate that, in fact, $\langle A\rangle = \langle B\rangle$ is solvable, the elements of $A$ and $B$ are $p$-elements for some prime $p$, and $\langle A\rangle $ is $p$-nilpotent. Moreover, under the second assumption, it turns out that $A=B$ and this is the only possible case. This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.
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 $G$.
In this paper it is proved that the group $F\left(\frac32\right)$, a Thompson-style group with breaks in $\mathbb{Z}\left[\frac16\right]$ but whose slopes are restricted only to powers of $\frac32$, is finitely generated, …
In this paper it is proved that the group $F\left(\frac32\right)$, a Thompson-style group with breaks in $\mathbb{Z}\left[\frac16\right]$ but whose slopes are restricted only to powers of $\frac32$, is finitely generated, with a generating set of two elements.
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.
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.
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of …
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes of these classes, respectively. These graphs are subgraphs of the ordinary ones associated to the conjugacy classes ofG, ${\Gamma}(G)$ and ${\Gamma}^{\ast}(G)$, which have been widely studied by several authors. We prove that the number of connected components of both graphs is at most 2, we determine the best upper bounds for the diameters and characterize the structure of $N$ when these graphs are disconnected.
Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the graph $\Gamma_G(N)$, which is the graph associated to the …
Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the graph $\Gamma_G(N)$, which is the graph associated to the conjugacy classes of $G$ contained in $N$, has no triangles and when the graph consists in exactly one triangle.
We prove that if a finite group $G$ contains a conjugacy class $K$ whose square is of the form $1 \cup D$, where $D$ is a conjugacy class of $G$, …
We prove that if a finite group $G$ contains a conjugacy class $K$ whose square is of the form $1 \cup D$, where $D$ is a conjugacy class of $G$, then $\langle K \rangle$ is a solvable proper normal subgroup of $G$ and we completely determine its structure. We also obtain the structure of those groups in which the assumption above is true for all non-central conjugacy classes and when every conjugacy class satisfies that its square is the union of all central conjugacy classes except at most one.
Suppose that $G$ is a finite group and $K$ a non-trivial conjugacy class of $G$ such that $KK^{-1}=1\cup D\cup D^{-1}$ with $D$ a conjugacy class of $G$. We prove that …
Suppose that $G$ is a finite group and $K$ a non-trivial conjugacy class of $G$ such that $KK^{-1}=1\cup D\cup D^{-1}$ with $D$ a conjugacy class of $G$. We prove that $G$ is not a non-abelian simple group. We also give arithmetical conditions on the class sizes determining the structure of $\langle K\rangle$ and $\langle D\rangle$. Furthermore, if $D=K$ is a non-real class, then $\langle K\rangle$ is $p$-elementary abelian for some odd prime $p$.
Many results have been established that show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this …
Many results have been established that show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper is to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character table of the group.
Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, …
Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there exists only a finite number of finite groups $G$, up to isomorphism, having a normal subgroup $N$ of index $n$ which contains exactly $s$ non-central $G$-conjugacy classes. We provide upper bounds for the orders of $G$ and $N$, which are used by using GAP to classify all finite groups with normal subgroups having a small index and few $G$-classes. We also study the corresponding problems when we only take into account the set of $G$-classes of prime-power order elements contained in a normal subgroup.
Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes …
Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, is equal to three.
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. …
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.
Gcharacter tables of a finite group G were defined before. These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of …
Gcharacter tables of a finite group G were defined before. These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of G. We analyze certain structural properties of normal subgroups which can be determined using their Gcharacter tables. For instance, we prove an extension of the Thompsons theorem from minimal Ginvariant characters of a normal subgroup. We also obtain a variation of Taketas theorem for hypercentral normal subgroups considering their minimal G-invariant characters. This generalization allows us to introduce a new class of nilpotent groups, the class of nMIgroups, whose members verify that its nilpotency class is bounded by the number of irreducible character degrees of the group.
Abstract A theorem of Z. Arad and E. Fisman establishes that if A and B are two non-trivial conjugacy classes of a finite group G such that either $$AB=A\cup B$$ …
Abstract A theorem of Z. Arad and E. Fisman establishes that if A and B are two non-trivial conjugacy classes of a finite group G such that either $$AB=A\cup B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> or $$AB=A^{-1} \cup B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>∪</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> , then G cannot be a non-abelian simple group. We demonstrate that, in fact, $$\langle A\rangle = \langle B\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>A</mml:mi> <mml:mo>⟩</mml:mo> <mml:mo>=</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>B</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> is solvable, the elements of A and B are p -elements for some prime p , and $$\langle A\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>A</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> is p -nilpotent. Moreover, under the second assumption, it turns out that $$A=B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> . This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.
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.
Abstract Let G be a finite group and let be the conjugacy class of an element x of G . In this paper, it is proved that if N is …
Abstract Let G be a finite group and let be the conjugacy class of an element x of G . In this paper, it is proved that if N is a normal subgroup of G such that the coset is the union of K and (the conjugacy class of the inverse of x ), then N and the subgroup are solvable. As an application, we prove that if there exists a natural number such that , then is solvable.
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. …
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.
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.
Products of conjugacy classes is a well-established theme in Group Theory with open conjectures. We summarize known and new results concerning the influence that the product of two conjugacy classes …
Products of conjugacy classes is a well-established theme in Group Theory with open conjectures. We summarize known and new results concerning the influence that the product of two conjugacy classes exerts on the structure of a finite group. We add several open questions in order to inspire the reader to solve them and develop new techniques of research.
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.
Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellon, who is financially supported by a predoctoral grant …
Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellon, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B-2015-77.
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.
Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see …
Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see text]. We show that, for any positive integers [Formula: see text] and [Formula: see text], there exist finitely many finite groups [Formula: see text], up to isomorphism, having a normal subgroup [Formula: see text] of index [Formula: see text] which contains exactly [Formula: see text] non-central [Formula: see text]-conjugacy classes. Upper bounds for the orders of [Formula: see text] and [Formula: see text] are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few [Formula: see text]-classes. We also study the related problems when we consider only the set of [Formula: see text]-classes of prime-power order elements contained in a normal subgroup.
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ . We determine the structure of $N$ when the diameter of the graph associated to …
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ . We determine the structure of $N$ when the diameter of the graph associated to the $G$ -conjugacy classes contained in $N$ is as large as possible, that is, equal to three.
Let N be a normal subgroup of a finite group G. In the recent past years some results have appeared concerning the influence of the G-class sizes of N, that …
Let N be a normal subgroup of a finite group G. In the recent past years some results have appeared concerning the influence of the G-class sizes of N, that is, with the sizes of the conjugacy classes in G contained in N, on the structure of N. In this survey, we present the main results and techniques used for proving that any normal subgroup of G which has exactly three G-conjugacy class sizes is solvable. Thus, we obtain a generalisation for normal subgroups of the classical N. Itô's theorem which asserts that those finite groups having three class sizes are solvable, and in particular, a new proof of it is provided.
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.
Abstract Let N be a normal subgroup of a group G and let p be a prime. We prove that if the p -part of | x | G is …
Abstract Let N be a normal subgroup of a group G and let p be a prime. We prove that if the p -part of | x | G is a constant for every prime-power order element x ∈ N ∖Z( N ), then N is solvable and has normal p -complement.
We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes.We show that if the set of G-class sizes of N …
We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes.We show that if the set of G-class sizes of N is {1, m, mp a }, with p a prime not dividing m, then N is solvable.Thus, we get a partial extension for normal subgroups on N. Itô's theorem on the solvability of groups having exactly three class sizes.
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups which can easily be verified from the character table.
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups which can easily be verified from the character table.
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups which can easily be verified from the character table.
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups which can easily be verified from the character table.
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).
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 ).
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.
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 .
Abstract Let G be a finite group. We prove that if the set of p -regular conjugacy class sizes of G has exactly two elements, then G has Abelian p …
Abstract Let G be a finite group. We prove that if the set of p -regular conjugacy class sizes of G has exactly two elements, then G has Abelian p -complement or G = PQ × A , with P ∈ Syl p ( G ), Q ∈ Syl q ( G ) and A Abelian.
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.
Let G be a finite p -solvable group for a fixed prime p . We determine the structure of G when the set of p -regular conjugacy class sizes of …
Let G be a finite p -solvable group for a fixed prime p . We determine the structure of G when the set of p -regular conjugacy class sizes of G is {1, m } for an arbitrary integer m > 1.
In 1953 N. Itô defined the conjugate rank of a finite group as the number of distinct sizes, not equal to 1, of the conjugacy classes of the group [7].
In 1953 N. Itô defined the conjugate rank of a finite group as the number of distinct sizes, not equal to 1, of the conjugacy classes of the group [7].
Let G be a finite group. We show that when the conjugacy class sizes of G are {1, m, n, mn}, with m and n positive integers such that (m, …
Let G be a finite group. We show that when the conjugacy class sizes of G are {1, m, n, mn}, with m and n positive integers such that (m, n) = 1, then G is solvable. As a consequence, we obtain that G is nilpotent and that m = pa and n = qb for two primes p and q.
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.
Let $G$ be a finite group and let $K$ be the conjugacy class of $x \in G$. If $K^2$ is a conjugacy class of $G$, then $[x,G]$ is solvable. If …
Let $G$ be a finite group and let $K$ be the conjugacy class of $x \in G$. If $K^2$ is a conjugacy class of $G$, then $[x,G]$ is solvable. If the order of $x$ is a power of prime, then $[x,G]$ has a normal $p$-complement. We also prove some related results on the solvability of certain normal subgroups when a non-trivial coset has certain properties.
Abstract Let G be a finite p -solvable group for a fixed prime p . We study how certain arithmetical conditions on the set of p -regular conjugacy class sizes …
Abstract Let G be a finite p -solvable group for a fixed prime p . We study how certain arithmetical conditions on the set of p -regular conjugacy class sizes of G influence the p -structure of G . In particular, the structure of the p -complements of G is described when this set is {1, m, n } for arbitrary coprime integers m, n > 1. The structure of G is determined when the noncentral p -regular class lengths are consecutive numbers and when all of them are prime powers.
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;
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 co-prime. Denote by n(Γ) the number of the connected components of Γ. By [1], n(Γ) ⩽ 2 for all finite groups, and if Γ is connected, the diameter of the graph is at most 4. In this paper we prove that if Γ is connected, then the diameter of the graph is at most 3, and this bound is the best possible. Similar results are proved for infinite FC-groups.
Let $M(G)$ be the subgroup of $G$ generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if $G$ is nilpotent, then …
Let $M(G)$ be the subgroup of $G$ generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if $G$ is nilpotent, then $M(G)$ has nilpotence class at most $3$. Using a slight variation on Mannâs methods, we obtain results that do not require us to assume that $G$ is nilpotent. We show that if $G$ is supersolvable, then $M(G)$ is nilpotent with class at most $3$, and in general, the Fitting subgroup of $M(G)$ has class at most $4$.
Abstract Let G be a finite group. For a ∈ G, let a G = {a g | g ∈ G} be the conjugacy class of a in G. In …
Abstract Let G be a finite group. For a ∈ G, let a G = {a g | g ∈ G} be the conjugacy class of a in G. In this paper, we study a conjecture due to Arad and Herzog which asserts that in a finite non-abelian simple group the product of two nontrivial conjugacy classes is never a single conjugacy class. In particular, we will verify this conjecture for several families of finite simple groups of Lie type.
We give a structure theorem for the finite groups with three conjugacy class sizes. In particular, they are solvable groups with derived length at most 3 or nilpotent groups.
We give a structure theorem for the finite groups with three conjugacy class sizes. In particular, they are solvable groups with derived length at most 3 or nilpotent groups.
It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G . The purpose of this article …
It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G . The purpose of this article is to give some analogous properties between these basic concepts.
We consider finite groups in which every triple of distinct conjugacy class sizes greater than one has a pair which is coprime. We prove such a group is soluble and …
We consider finite groups in which every triple of distinct conjugacy class sizes greater than one has a pair which is coprime. We prove such a group is soluble and has conjugate rank at most three.
IntroductionMuch attention has recently been given to the characterization of classes of simple groups in terms of conditions which specify the centralizers of their revolutions or their Sylow 2-subgroups.(Cf.R. Brauer …
IntroductionMuch attention has recently been given to the characterization of classes of simple groups in terms of conditions which specify the centralizers of their revolutions or their Sylow 2-subgroups.(Cf.R. Brauer [2]; R. Brauer, M.Suzuki, and G
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.
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).
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.
We survey the problem of constructing the groups of a given finite order. We provide an extensive bibliography and outline practical algorithmic solutions to the problem. Motivated by the millennium, …
We survey the problem of constructing the groups of a given finite order. We provide an extensive bibliography and outline practical algorithmic solutions to the problem. Motivated by the millennium, we used these methods to construct the groups of order at most 2000; we report on this calculation and describe the resulting group library.