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.
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.
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.
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.
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. $G=O_p(G) \times O_{p'}(G)$.
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. $G=O_p(G) \times O_{p'}(G)$.
We determine the structure of finite $\pi(m)$-separable groups if the set of conjugacy class sizes of primary and biprimary elements is $\{1, m, mn\}$, where $m$ and $n$ are two …
We determine the structure of finite $\pi(m)$-separable groups if the set of conjugacy class sizes of primary and biprimary elements is $\{1, m, mn\}$, where $m$ and $n$ are two coprime integers.
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.
In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set …
In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of order $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any $n\in \mathbb{N}$, solvable groups of $n^\text{th}$-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or $A_5$ and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.
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.
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.
Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes …
Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We prove sharp lower bounds for this number in terms of $p$ and the $p'$-part of the order of $G$ which ensure that $G$ is $p$-solvable. A bound for the $p$-length is obtained which is sharp for odd primes $p$. We also prove a new best possible criterion for the existence of a normal Sylow $p$-subgroup in terms of these quantities.
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.
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.
The (undirected) power graph on the conjugacy classes $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct …
The (undirected) power graph on the conjugacy classes $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent in $mathcal{P_C}(G)$ if one is a subset of a power of the other. In this paper, we describe groups whose associated graphs are $k$-regular for $k=5,6$.
Let $G$ be a finite group, $p$ a prime, and $IBr_p(G)$ the set of irreducible $p$-Brauer characters of $G$. Let $\bar e_p(G)$ be the largest integer such that $p^{\bar e_p(G)}$ …
Let $G$ be a finite group, $p$ a prime, and $IBr_p(G)$ the set of irreducible $p$-Brauer characters of $G$. Let $\bar e_p(G)$ be the largest integer such that $p^{\bar e_p(G)}$ divides $\chi(1)$ for some $\chi \in IBr_p(G)$. We show that $|G:O_p(G)|_p \leq p^{k \bar e_p(G)}$ for an explicitly given constant $k$. We also study the analogous problem for the $p$-parts of the conjugacy class sizes of $p$-regular elements of finite groups.
Let $G$ be a finite group and $cd(G)$ denote the character degree set for $G$. The prime graph $DG$ is a simple graph whose vertex set consists of prime divisors …
Let $G$ be a finite group and $cd(G)$ denote the character degree set for $G$. The prime graph $DG$ is a simple graph whose vertex set consists of prime divisors of elements in $cd(G)$, denoted $rho(G)$. Two primes $p,qin rho(G)$ are adjacent in $DG$ if and only if $pq|a$ for some $ain cd(G)$. We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.
Let G be a finite solvable group and p a prime \neq 2 .The purpose of this note is to give the structure of finite solvable groups with exactly four …
Let G be a finite solvable group and p a prime \neq 2 .The purpose of this note is to give the structure of finite solvable groups with exactly four p-regular conjugacy classes.
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.
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.
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.
One of the important invariants of a р -block B of a group algebra is ℓ ( B ), the number of non-isomorphic simple B-modules. A number of authors calculated …
One of the important invariants of a р -block B of a group algebra is ℓ ( B ), the number of non-isomorphic simple B-modules. A number of authors calculated ℓ (B) for various types of defect groups of B. In particular, by Olsson [6], it has been proved that if p = 2 and the defect groups of the block B are dihedral or semi-dihedral or generalized quaternion, then ℓ (B) is at most 3. In this paper, we restrict our attention to the principal p -block B 0 of a finite р -solvable group with ℓ (B 0 ) ≤ 3. Let Γ be a finite р -solvable group and k a splitting field for Γ with characteristic р .