Search results for "Finite group"
showing 10 items of 205 documents
Criteria for the solubility and non-simplicity of finite groups
2005
Abstract Some criteria of the non-simplicity of a finite group by graph theoretical terms are derived. This is then used to establish conditions under which a finite group is soluble.
On Formations of Finite Groups with the Wielandt Property for Residuals
2001
Abstract Given two subgroups U, V of a finite group which are subnormal subgroups of their join 〈U, V〉 and a formation F , in general it is not true that 〈U, V〉 F = 〈U F , V F 〉. A formation is said to have the Wielandt property if this equality holds universally. A formation with the Wielandt property must be a Fitting class. Wielandt proved that the most usual Fitting formations (e.g., nilpotent groups and π-groups) have the Wielandt property. At present, neither a general satisfactory result on the universal validity of the Wielandt property nor a counterexample is known. In this paper a criterion for a Fitting formation to have the Wielandt property is given. As an application, it is p…
SUBGROUPS OF FINITE GROUPS WITH A STRONG COVER-AVOIDANCE PROPERTY
2009
AbstractA subgroup A of a group G has the strong cover-avoidance property in G, or A is a strong CAP-subgroup of G, if A either covers or avoids every chief factor of every subgroup of G containing A. The main aim of the present paper is to analyse the impact of the strong cover and avoidance property of the members of some relevant families of subgroups on the structure of a group.
Real groups and Sylow 2-subgroups
2016
Abstract If G is a finite real group and P ∈ Syl 2 ( G ) , then P / P ′ is elementary abelian. This confirms a conjecture of Roderick Gow. In fact, we prove a much stronger result that implies Gow's conjecture.
p-Length andp′-Degree Irreducible Characters Having Values in ℚp
2013
Let G be a p-solvable group of p-length l, where p is any prime. We show that G has at least 2 l irreducible characters of degree coprime to p and having values inside ℚ p . This generalizes a previous result for p = 2 [6] to arbitrary primes. With the same notation, we prove that if p is odd then G has at least 2 l Galois orbits of conjugacy classes of p-elements having values in ℚ p .
Character sums and double cosets
2008
Abstract If G is a p-solvable finite group, P is a self-normalizing Sylow p-subgroup of G with derived subgroup P ′ , and Ψ is the sum of all the irreducible characters of G of degree not divisible by p, then we prove that the integer Ψ ( P ′ z P ′ ) is divisible by | P | for all z ∈ G . This answers a question of J. Alperin.
McKay natural correspondences on characters
2014
Let [math] be a finite group, let [math] be an odd prime, and let [math] . If [math] , then there is a canonical correspondence between the irreducible complex characters of [math] of degree not divisible by [math] belonging to the principal block of [math] and the linear characters of [math] . As a consequence, we give a characterization of finite groups that possess a self-normalizing Sylow [math] -subgroup or a [math] -decomposable Sylow normalizer.
Quadratic rational solvable groups
2012
Abstract A finite group G is quadratic rational if all its irreducible characters are either rational or quadratic. If G is a quadratic rational solvable group, we show that the prime divisors of | G | lie in { 2 , 3 , 5 , 7 , 13 } , and no prime can be removed from this list. More generally, if G is solvable and the field Q ( χ ) generated by the values of χ over Q satisfies | Q ( χ ) : Q | ⩽ k , for all χ ∈ Irr ( G ) , then the set of prime divisors of | G | is bounded in terms of k . Also, we prove that the degree of the field generated by the values of all characters of a semi-rational solvable group (see Chillag and Dolfi, 2010 [1] ) or a quadratic rational solvable group over Q is bou…
Finite groups with some C-normal minimal subgroups
2000
Abstract Let G be a finite group. The question of how the properties of its minimal subgroups influence the structure of G is of considerable interest for some scholars. Several authors have investigated this question by using normal or quasinormal conditions. In this paper we use c -normal condition on minimal subgroups to characterize the structure of G through the theory of formations.
A reduction theorem for a conjecture on products of two π -decomposable groups
2013
[EN] For a set of primes pi, a group X is said to be pi-decomposable if X = X-pi x X-pi' is the direct product of a pi-subgroup X-pi and a pi'-subgroup X-pi', where pi' is the complementary of pi in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: "Let pi be a set of odd primes. If the finite group G = AB is a product of two pi-decomposable subgroups A = A(pi) x A(pi') and B = B-pi x B-pi', then A(pi)B(pi) = B(pi)A(pi) and this is a Hall pi-subgroup of G." We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper. (C) 2013 Elsevier Inc. All ri…