Solvable Groups

Consider G to be a finite group and p to be a prime divisor of the order |G| in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro's vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let φ ∈ Irr(G) have a positive height, and let there be an anchor group A φ. We prove that if the normalizer N G (A φ) is p-constrained, then O ṕ(N G (A φ)) = {1 G }, where O ṕ(N G (A φ)) is the maximal normal ṕ subgroup of N G (A φ). We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results.

We call a subalgebra U of a Lie algebra L a CAP -subalgebra of L if for any chief factor H/K of L, we have In this paper we investigate some properties of such subalgebras and obtain some characterizations for a finite-dimensional Lie algebra L to be solvable under the assumption that some of its maximal subalgebras or 2-maximal subalgebras be CAP -subalgebras.

Questo lavoro è rilasciato con licenza Creative Commons Attribution 4.0 International (CC BY 4.0). Viene consentita la riproduzione, distribuzione, comunicazione pubblica e modifica, anche per fini commerciali, a condizione di attribuire la paternità dell'opera all'autore. Maggiori informazioni sulla licenza sono disponibili al seguente link: Per il livello avanzato di questo lavoro, si richiede una solida e rigorosa conoscenza dei seguenti argomenti: i. Teoremi di base: principio di sostituzione di Steinitz, rango-nullità, teorema di Cayley-Hamilton, polinomio caratteristico e polinomio minimale, decomposizione di Jordan-Chevalley, diagonalizzabilità e semisemplicità. ii. Traccia e determinante: compatibilità con composizione e restrizione di scalari. Forma di traccia ⟨x, y⟩ = Tr(xy) su estensioni finite. Non-degenerazione della forma di traccia per estensioni separabili e discriminante di una base. iii. Tensoriale: basi tensori e comportamento di dimensione dim Criteri di iniettività e suriettività su tensori di mappe lineari. i. Azioni di gruppi su insiemi: orbita-stabilizzatore, equazione di classe, trasitività e primitività, blocchi. Azioni su radici di polinomi e su coperture. ii. Struttura di gruppi finiti: teoremi di Cauchy e Sylow, serie di Jordan-Hölder, gruppi risolubili e serie derivata. Prodotti semi-diretti, presentazioni e gruppi diedrali, simmetrici, alterni. iii. Profinito: limiti inversi di gruppi finiti, topologia di Krull e compattezza di Tychonoff. Sottogruppi aperti e quozienti finiti. i. Anelli e moduli: localizzazione, lemma di Nakayama, teorema cinese del resto. Integralità e chiusura integrale, domini UFD, PID e di Dedekind, proprietà di fattorizzazione unica degli ideali. ii. Polinomi: gradicazione, divisione, fattorizzazione in campi e domini, criteri di irriducibilità, lemma di Gauss, criterio di Schönemann-Eisenstein. Risultante e discriminante, criterio di separabilità gcd(f, f ′ ) = 1. iii. Campi: elementi algebrici, polinomio minimale e grado. Separabilità, normalità, campi di spezzamento e chiusure algebriche. i. Campi di numeri: anello degli interi O K e sue proprietà, ideali frazionari, differente e discriminante relativo, forma di traccia. ii. Completamenti: DVR, valutazioni normalizzate, completamenti K p , campi residui e uniformizzanti. Teorema di Hensel. iii. Decomposizione dei primi: indici di ramificazione e, gradi di inerzia f , criteri elementari di ramificazione e non-ramificazione.

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let f cf,sol (n) denote the isomorphism classes of solvable cubefree groups of order n. We find asymptotic bounds for f cf,sol (n) in this paper. Let p be a prime and let q = p k for some positive integer k. We also give a formula for the number of conjugacy classes of the subgroups that are maximal amongst non-abelian solvable cube-free p ′-subgroups of GL(2, q). Further, we find the exact number of split extensions of P by Q up to isomorphism of a given order where P ∈ {Z p ×Z p , Z p α }, p is a prime, α is a positive integer and Q is a cube-free abelian group of odd order such that p ∤ |Q|.

The symmetric maximum, denoted by , is an extension of the usual maximum ∨ operation so that 0 is the neutral element, and -x is the symmetric (or inverse) of x, i.e., x (-x) = 0. However, such an extension does not preserve the associativity of ∨. This fact asks for systematic ways of parenthesing (or bracketing) terms of a sequence (with more than two arguments) when using such an extended maximum. We refer to such systematic (predefined) ways of parenthesing as computation rules. As it turns out there are infinitely many computation rules, each of which corresponds to a systematic way of bracketing arguments of sequences. Essentially, computation rules reduce to deleting terms of sequences based on the condition x (-x) = 0. This observation gives raise to a quasi-order on the set of such computation rules: say that rule 1 is below rule 2 if for all sequences of numbers, rule 1 deletes more terms of the sequence than rule 2. In this paper we present a study of this quasi-ordering of computation rules. In particular, we show that the induced poset of all equivalence classes of computation rules is uncountably infinite, has infinitely many maximal elements, has infinitely many atoms, and it embeds the powerset of natural numbers ordered by inclusion.

La teoria di Galois rappresenta uno dei vertici più straordinari della matematica pura, un capolavoro di astrazione e di potenza concettuale che ha rivoluzionato la nostra comprensione delle equazioni algebriche e delle simmetrie matematiche...

La presente trattazione è dedicata ai fasci di circonferenze, ovvero un insieme di circonferenze i cui centri giacciono su una retta, o anche un insieme di circonferenze aventi il medesimo centro ed è altresì dedicata alle circonferenze di Apollonio inserite nel contesto dei fasci di circonferenze e di cui si darà una genesi geometrica. Nella trattazione è inoltre introdotto un luogo geometrico caratteristico di diversi fasci di circonferenze rappresentato da un'iperbole equilatera; di tale luogo geometrico verranno indicate alcune sue proprietà.

Let π(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G, denoted Γ G , is the graph with vertex set π(G) with edges {p, q} ∈ E(Γ G ) if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with minimal prime graphs, and we show that solvable groups whose prime graphs are minimal have Fitting length 3 or 4.

We give a shorter proof of the following theorem of Kathryn Mann [M]: the identity component of the group of the compactly supported C diffeomorphisms of R cannot admit a nontrivial C-action on S, provided n ≥ 2, r 6= n+ 1 and p ≥ 2. We also give a new proof of another theorem of Mann [M]: any nontrivial homomorphism from the group of the orientation preserving C diffeomorphisms of the circle to the group of C diffeomorphisms of the circle is the conjugation of the standard inclusion by a C diffeomorphism, if r ≥ p, r 6= 2 and p 6= 1.

It is an interesting topic to determine the structure of a finite group which has a given number of elements of maximal order. In this article, the author classified finite groups with 30 elements of maximal order. Keywords A finite group • The number of elements of maximal order • Classification • Thompson's conjecture Mathematics Subject Classifications (2000) 20E34 • 20E45

For a group G, we define a graph ∆(G) by letting G # = G\{ 1 } be the set of vertices and by drawing an edge between distinct elements x, y ∈ G # if and only if the subgroup x, y is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate ∆(G) for a Z-group G.

For a subgroup L of the symmetric group S ℓ , we determine the minimal base size of GL d (q) ≀ L acting on V d (q) ℓ as an imprimitive linear group. This is achieved by computing the number of orbits of GL d (q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of Vm(q). We then use these results to prove that for certain families of subgroups L, the affine groups whose stabilisers are large subgroups of GL d (q) ≀ L satisfy a conjecture of Pyber concerning bases.

We study a family of metrics on Euclidean space that generalize the left-invariant metric of the SOL group and the metric of the logarithmic model of Hyperbolic space. Suppose G is a connected, simply-connected, Heintze group of Abelian type with diagonalizable derivation or the horospherical product of two such groups. In this scenario, G is isometric to Euclidean space with a metric of the type considered. We have derived a formula for the volume entropy of metrics in this family and used it to solve a conjecture related to a family of 3-manifolds that interpolates between the SOL group and hyperbolic space.

First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra. We give an isomorphic characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras. Next, we find a Jordan-admissible condition for a Novikov algebra $\N$. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403-453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.

This paper involves an investigation on the maximal subgroups of the groups of order pqr, a theorem has been stated and prove concerning the nature and behavior of the maximal subgroup of such groups. The group algorithm and programming (GAP) has been applied to enhance and validate the results.

Let G be a finite group, and write cd(G) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if, for any distinct degrees a, b ∈ cd(G), the total number of (not necessarily different) primes of the greatest common divisor gcd(a, b) is at most 2. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL 2 (q) for q ≥ 7.

In this paper, we consider lifts of π-partial characters with the property that the irreducible constituents of their restrictions to certain normal subgroups are also lifts. We will show that such a lift must be induced from what we call an inductive pair, and every character induced from an inductive pair is a such a lift. With this condition, we will get a lower bound on the number of such lifts.

In this paper, we show that if p is a prime and G is a psolvable group, then |G : Op(G)|p ≤ (b(G) p /p) 1/(p−1) where b(G) is the largest character degree of G. If p is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow p-subgroup of G is at most p, then |G : Op(G)|p ≤ b(G).

Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if G is an M-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if G is an M-group of odd order where every irreducible character is a {p}-lift for some prime p. We say that a group where irreducible character is super monomial is a super M-group. We use our results to find an example of a super M-group that has a subgroup that is not a super M-group. MSC primary : 20C15

Let G be a solvable group. Let p be a prime and let Q be a p-subgroup of a subgroup V. Suppose ϕ ∈ IBr(G). If either |G| is odd or p = 2, we prove that the number of Brauer characters of H inducing ϕ with vertex Q is at most |N G (Q) : N V (Q)|.

In this paper we examine the behavior of lifts of Brauer characters in solvable groups. In the main result, we show that if ϕ ∈ IBr p (G) is a Brauer character of a solvable group such that ϕ has an abelian vertex subgroup Q, then the number of lifts of ϕ in Irr(G) is at most |Q|. In order to accomplish this, we develop several results about lifts of Brauer characters in solvable groups that were previously only known to be true in the case of groups of odd order.

We investigate character degree graphs of solvable groups. In particular, we provide general results that can be used to eliminate which degree graphs can occur as solvable groups. Finally, we show a specific family of graphs cannot occur as a character degree for any solvable group.

For a Lie algebra L and a subalgebra M of L we say that a subalgebra U of L is a supplement to M in L if L = M + U . We investigate those Lie algebras all of whose maximal subalgebras have abelian supplements, those that have nilpotent supplements, those that have nil supplements, and those that have supplements with the property that their derived algebra is inside the maximal subalgebra being supplemented. For the algebras over an algebraically closed field of characteristic zero in the last three of these classes we find complete descriptions; for those in the first class partial results are obtained.

A finite-dimensional Lie algebra L over a field F of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in .

Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M . The set I(M ) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular finding new characterisations of solvable and supersolvable Lie algebras.

We describe under various conditions abelian subgroups of the automorphism group Aut(Tn) of the regular n-ary tree Tn, which are normalized by the n-ary adding machine τ = (e,. .. , e, τ)στ where στ is the n-cycle (0, 1,. .. , n − 1). As an application, for n = p a prime number, and for n = 4, we prove that every soluble subgroup of Aut(Tn), containing τ is an extension of a torsion-free metabelian group by a finite group.

Let G be a finite group, Irr(G) be the set of irreducible characters of G, and denote by cd(G), the set of irreducible character degrees of G. The character degree graph of G, which is denoted by Γ(G), is defined as follows: the vertices of this graph are the prime divisors of the irreducible character degrees of the group G and two distinct vertices p1 and p2 are joined by an edge if there exists an irreducible character degree of G which is divisible by p1p2. This graph was introduced in [13] and studied by many authors (see [11, 14]). Recently, there has been much interest in the influence of arithmetical conditions on degrees of irreducible characters of a group G on the structure of G. A finite group G is called a K3-group if |G| has exactly three distinct prime divisors. Recently Chen et al. in [15] and [16] proved that all simple K3-groups and the Mathieu groups are uniquely determined by their orders and one or both of their largest and second largest irreducible character d...

In this paper, we used wreath products of two permutation groups  in constructing transitive supersolvable permutation groups. We verified these groups using some groups theoretical concepts and also validate our work using a standard program; GAP (Groups Algorithm and Programming).

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This paper is part of a program to study the conjecture of E.C. Dade on counting characters in blocks for several finite groups. In this paper, we verify Dade's invariant conjecture for the Chevalley groups G 2 (p a) in the defining characteristic when p = 2 or 3. This implies Dade's final conjecture when p = 2 or 3.

Let A be a commutative algebra over a field F of characteristic = 2, 3. In [M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J. 27 (1960) 21-31], M. Gerstenhaber proved that if A is a nilalgebra of bounded index t and the characteristic of F is zero (or greater than 2t − 3), then the right multiplication R x

Let A be a commutative power-associative nilalgebra. In this paper we prove that when A (of characteristic = 2) is of dimension ≤ 10 and the identity x 4 = 0 is valid in A, then ((y 2)x 2)x 2 = 0 for all y, x in A and ((A 2) 2) 2 = 0. That is, A is solvable.

Let A be a commutative algebra over a field F of characteristic = 2, 3. In [M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J. 27 (1960) 21-31], M. Gerstenhaber proved that if A is a nilalgebra of bounded index t and the characteristic of F is zero (or greater than 2t − 3), then the right multiplication R x

Let α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.