Search results for "character"
showing 10 items of 2956 documents
Large characteristically simple sections of finite groups
2021
In this paper we prove that if G is a group for which there are k non-Frattini chief factors isomorphic to a characteristically simple group A, then G has a normal section C/R that is the direct product of k minimal normal subgroups of G/R isomorphic to A. This is a significant extension of the notion of crown for isomorphic chief factors.
Permutable subnormal subgroups of finite groups
2009
The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called PT -groups. In particular, it is shown that the finite solvable PT -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugatepermutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalentl…
An answer to a question of Isaacs on character degree graphs
2006
Abstract Let N be a normal subgroup of a finite group G. We consider the graph Γ ( G | N ) whose vertices are the prime divisors of the degrees of the irreducible characters of G whose kernel does not contain N and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of G whose kernel does not contain N. We prove that if Γ ( G | N ) is disconnected then G / N is solvable. This proves a strong form of a conjecture of Isaacs.
An answer to two questions of Brewster and Yeh on M-groups
2003
Let χ be a (complex) irreducible character of a finite group. Recall that χ is monomial if there exists a linear character λ ∈ Irr(H), where H is some subgroup of G, such that χ = λG. A group is an M -group if all its irreducible characters are monomial. In 1992, B. Brewster and G. Yeh [1] raised the following two questions. Question A. Let M and N be normal subgroups of a group G. Assume that (|G : M |, |G : N |) = 1 and that M and N are M -groups. Does this imply that G is an M -group? ∗Research supported by the Basque Government, the Spanish Ministerio de Ciencia y Tecnoloǵia and the University of the Basque Country
On maximal subgroups of finite groups
1991
(1991). On maximal subgroups of finite groups. Communications in Algebra: Vol. 19, No. 8, pp. 2373-2394.
A characteristic subgroup and kernels of Brauer characters
2005
If G is finite group and P is a Sylow p-subgroup of G, we prove that there is a unique largest normal subgroup L of G such that L ⋂ P = L ⋂ NG (P). If G is p-solvable, then L is the intersection of the kernels of the irreducible Brauer characters of G of degree not divisible by p.
On the product of a nilpotent group and a group with non-trivial center
2007
Abstract It is proved that a finite group G = A B which is a product of a nilpotent subgroup A and a subgroup B with non-trivial center contains a non-trivial abelian normal subgroup.
Characters of relative p'-degree over normal subgroups
2013
Let Z be a normal subgroup of a finite group G , let ??Irr(Z) be an irreducible complex character of Z , and let p be a prime number. If p does not divide the integers ?(1)/?(1) for all ??Irr(G) lying over ? , then we prove that the Sylow p -subgroups of G/Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture
A Characterization of the Class of Finite Groups with Nilpotent Derived Subgroup
2002
The class of all finite groups with nilpotent commutator subgroup is characterized as the largest subgroup-closed saturated formation 𝔉 for which the 𝔉-residual of a group generated by two 𝔉-subnormal subgroups is the subgroup generated by their 𝔉–residuals.
On fully ramified Brauer characters
2014
Let Z be a normal subgroup of a finite group, let p≠5 be a prime and let λ∈IBr(Z) be an irreducible G-invariant p-Brauer character of Z. Suppose that λG=eφ for some φ∈IBr(G). Then G/Z is solvable. In other words, a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group.