Search results for "Groups"
showing 10 items of 1120 documents
The minimal number of characters over a normal p-subgroup
2007
Abstract If N is a normal p-subgroup of a finite group G and θ ∈ Irr ( N ) is a G-invariant irreducible character of N, then the number | Irr ( G | θ ) | of irreducible characters of G over θ is always greater than or equal to the number k p ′ ( G / N ) of conjugacy classes of G / N consisting of p ′ -elements. In this paper, we investigate when there is equality.
Real constituents of permutation characters
2022
Abstract We prove a broad generalization of a theorem of W. Burnside about the existence of real characters of finite groups to permutation characters. If G is a finite group, under the necessary hypothesis of O 2 ′ ( G ) = G , we can also give some control on the parity of multiplicities of the constituents of permutation characters (a result that needs the Classification of Finite Simple Groups). Along the way, we give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11 which asks how much information about subgroups of a finite group can be determined by the character table.
The McKay conjecture and Galois automorphisms
2004
The main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group G can be computed locally. The simplest of these conjectures is the ?McKay conjecture? which asserts that the number of irreducible complex characters of G of degree not divisible by p is the same if computed in a p-Sylow normalizer of G. In this paper, we propose a much stronger version of this conjecture which deals with Galois automorphisms. In fact, the same idea can be applied to the celebrated Alperin and Dade conjectures.
Finite groups with subgroups supersoluble or subnormal
2009
Abstract The aim of this paper is to study the structure of finite groups whose non-subnormal subgroups lie in some subclasses of the class of finite supersoluble groups.
A Local Approach to Certain Classes of Finite Groups
2003
Abstract We develop several local approaches for the three classes of finite groups: T-groups (normality is a transitive relation) and PT-groups (permutability is a transitive relation) and PST-groups (S-permutability is a transitive relation). Here a subgroup of a finite group G is S-permutable if it permutes with all the Sylow subgroup of G.
Self-normalizing Sylow subgroups
2003
Using the classification of finite simple groups we prove the following statement: Let p > 3 p>3 be a prime, Q Q a group of automorphisms of p p -power order of a finite group G G , and P P a Q Q -invariant Sylow p p -subgroup of G G . If C N G ( P ) / P ( Q ) \mathbf {C}_{\mathbf {N}_G(P)/P}(Q) is trivial, then G G is solvable. An equivalent formulation is that if G G has a self-normalizing Sylow p p -subgroup with p > 3 p >3 a prime, then G G is solvable. We also investigate the possibilities when p = 3 p=3 .
Central idempotents and units in rational group algebras of alternating groups
1998
Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theory of Young tableaux, we give an explicit description of the minimal central idempotents of ℚAn. As an application we construct finitely many generators for a subgroup of finite index in the centre of the group of units of ℚAn.
$n$-th relative nilpotency degree and relative $n$-isoclinism classes
2011
P. Hall introduced the notion of isoclinism between two groups more than 60 years ago. Successively, many authors have extended such a notion in different contexts. The present paper deals with the notion of relative n-isoclinism, given by N. S. Hekster in 1986, and with the notion of n-th relative nilpotency degree, recently introduced in literature.
Groups whose real irreducible characters have degrees coprime to p
2012
Abstract In this paper we study groups for which every real irreducible character has degree not divisible by some given odd prime p .
The number of lifts of a Brauer character with a normal vertex
2011
AbstractIn this paper we examine the behavior of lifts of Brauer characters in p-solvable groups. In the main result, we show that if φ∈IBr(G) has a normal vertex Q and either p is odd or Q is abelian, then the number of lifts of φ is at most |Q:Q′|. As a corollary, we prove that if φ∈IBr(G) has an abelian vertex subgroup Q, then the number of lifts of φ in Irr(G) is at most |Q|.