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We prove several results giving substantial evidence in support of the conjectural existence of a Glauberman-Isaacs bijection for Brauer characters under a coprime action. We also discuss related bijections for the McKay conjecture. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that if p is an odd prime and G is a finite group satisfying the condition that p² divides the degree of no irreducible character of G, then |G : Op(G)|p ≤ p4, where Op(G) is the largest normal p-subgroup of G, and if P is a Sylow p-subgroup of G, then P" is subnormal in G. Our investigations suggest that if pa is the largest power of p dividing the degrees of irreducible characters of G, then |G : Op(G)|p is bounded by pf(a), where f(a) is a function in a and P(a+1) is subnormal in G. [ABSTRACT FROM AUTHOR]

Abstract: In a solvable group G, if does not divide for all , then we prove that . This bound is best possible. [Copyright &y& Elsevier]

Abstract: Let G be a finite group and let p be an odd prime. Under certain conditions on the p-parts of the degrees of its irreducible p-Brauer characters, we prove the solvability of G. As a consequence, we answer a question proposed by B. Huppert in 1991: If G has exactly two distinct irreducible p-Brauer character degrees, then is G solvable? We also determine the structure of non-solvable groups with exactly two irreducible 2-Brauer character degrees. [Copyright &y& Elsevier]

Suppose that G is a finite group and that p is a prime number. We prove that if every p-rational irreducible character of G is non-zero on every p-element of G, then G has a normal Sylow p-subgroup. This yields a p-rational refinement of the Itô–Michler theorem: if p does not divide the degree of any irreducible p-rational character of G, then G has a normal Sylow p-subgroup [ABSTRACT FROM PUBLISHER]

Abstract: Suppose that G is a finite group, let p be a prime and let . We prove that P is normal in G if and only if all the irreducible constituents of the permutation character have degree not divisible by p. [Copyright &y& Elsevier]

In this paper we study finite p-solvable groups having irreducible complex characters χ ∈ Irr(G) which take root of unity values on the p-singular elements of G. [ABSTRACT FROM AUTHOR]

Let 𝕂 be an algebraically closed field. We classify the finite groups having exactly one irreducible 𝕂-representation of degree bigger than one. The case where the characteristic of 𝕂 is zero, was done by G. Seitz in 1968. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in 1993, which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes. This allows us to introduce the fuzzy prime radical and contribute to establish the basis of a Fuzzy Noncommutative Ring Theory. [Copyright &y& Elsevier]

Abstract: In this paper we study groups for which every real irreducible character has degree not divisible by some given odd prime p. [Copyright &y& Elsevier]

Abstract: A classical theorem of John Thompson on character degrees states that if the degree of any complex irreducible character of a finite group G is 1 or divisible by a prime p, then G has a normal p-complement. In this paper, we consider fields of values of characters and prove some improvements of this result. [Copyright &y& Elsevier]

Abstract: Let G be a finite group, and let denote the set of irreducible complex characters of G. An element x of G is non-vanishing if, for every χ in , we have . We prove that, if x is a non-vanishing element of G and the order of x is coprime to 6, then x lies in the Fitting subgroup of G. [Copyright &y& Elsevier]

Abstract: We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes. [Copyright &y& Elsevier]

Abstract: We study the finite groups in which every irreducible rational valued character is linear, and those in which every rational element is central. [Copyright &y& Elsevier]

Abstract: We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed serial coalgebras are characterized. The Auslander–Reiten quiver of a serial coalgebra is described. Finally, a version of Eisenbud–Griffith Theorem is proved, namely, every subcoalgebra of a prime, hereditary and strictly quasi-finite coalgebra is serial. [Copyright &y& Elsevier]

In this paper, it is proved that if B is a Brauer p-block of a p-solvable group, for some odd prime p, then the height of any ordinary character in B is at most 2b, where pb is the largest degree of the irreducible characters of the defect group of B. Some other results that relate the heights of characters with properties of the defect group are obtained. 2000 Mathematics Subject Classification 20C15, 20C20. [ABSTRACT FROM PUBLISHER]

We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we examine the behavior of lifts of Brauer characters in p-solvable groups. In the main result, we show that if has a normal vertex Q and either p is odd or Q is abelian, then the number of lifts of φ is at most . As a corollary, we prove that if has an abelian vertex subgroup Q, then the number of lifts of φ in is at most . [ABSTRACT FROM AUTHOR]