Search results for "Character degrees"

showing 5 items of 5 documents

Groups with exactly one irreducible character of degree divisible byp

2014

Let [math] be a prime. We characterize those finite groups which have precisely one irreducible character of degree divisible by [math] .

AlgebraPure mathematicsAlgebra and Number TheoryCharacter (mathematics)character degreesCharacter tableDegree (graph theory)characters20C15Character groupfinite groupsMathematicsAlgebra & Number Theory
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Characterizing normal Sylow p-subgroups by character degrees

2012

Abstract Suppose that G is a finite group, let p be a prime and let P ∈ Syl p ( G ) . We prove that P is normal in G if and only if all the irreducible constituents of the permutation character ( 1 P ) G have degree not divisible by p.

Finite groupAlgebra and Number TheoryDegree (graph theory)010102 general mathematicsSylow theoremsPrimitive permutation group01 natural sciencesPrime (order theory)Characters of finite groupsCharacter degrees010101 applied mathematicsCombinatoricsPermutationCharacter (mathematics)0101 mathematicsMathematicsJournal of Algebra
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Finite groups with real-valued irreducible characters of prime degree

2008

Abstract In this paper we describe the structure of finite groups whose real-valued nonlinear irreducible characters have all prime degree. The more general situation in which the real-valued irreducible characters of a finite group have all squarefree degree is also considered.

Finite groupReal charactersBrauer's theorem on induced charactersAlgebra and Number TheoryDegree (graph theory)Mathematics::Number TheoryStructure (category theory)Prime elementSquare-free integerCharacter degreesCombinatoricsCharacter tableMathematics::Representation TheoryCharacter groupMathematicsJournal of Algebra
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Complex group algebras of finite groups: Brauer's Problem 1

2007

Abstract Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m . We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.

Mathematics(all)Modular representation theoryPure mathematicsFinite groupBrauer's Problem 1Group (mathematics)General MathematicsCharacter degreesCombinatoricsRepresentation theory of the symmetric groupGroup of Lie typeSymmetric groupSimple groupGroup algebraFinite groupRepresentation theory of finite groupsMathematicsAdvances in Mathematics
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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.

Normal subgroupCombinatoricsDiscrete mathematicsFinite groupMathematics(all)ConjectureGeneral MathematicsProjective charactersNormal subgroupsSolvable groupsCharacter degreesGraphMathematicsAdvances in Mathematics
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