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Let 퐺 be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type 햠. In this paper, we give an explicit description of the action of automorphisms of 퐺 on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. Späth [Equivariant character correspondences and inductive McKay condition for type 햠, J. Reine Angew. Math.728 (2017), 153–194] and provides a useful tool for investigating the local sides of the local-global conjectures as one usually needs to deal with Levi subgroups. As an application we obtain a generalization of the stabilizer condition in the so-called inductive McKay condition [B. Späth, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc.44 (2012), 3, 426–438; Theorem 2.12] for irreducible characters of 퐺. Moreover, a criterion is given to explicitly determine whether an irreducible character is a constituent of a given generalized Gelfand–Graev character of 퐺. [ABSTRACT FROM AUTHOR]

In this paper, the non-abelian simple groups having at most 20 distinct irreducible character degrees are classified. [ABSTRACT FROM AUTHOR]

A finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let cd (G) be the set of degrees of complex irreducible characters of G. An important program in character theory is to obtain the structure of G when cd (G) is known. In this article, we consider a dual version of this approach. Indeed, we consider non-solvable groups with five character degrees and present a description of the set of character degrees for such groups. As an application of this result, it is shown that a finite group with five character degrees and containing no prime power must be a solvable group. [ABSTRACT FROM AUTHOR]

The aim of this article is to contribute to a question of R. Brauer that "when do non-isomorphic groups have isomorphic complex group algebras?" Let H and G be finite groups where PSU n (q) ≤ G ≤ PGU n (q) , and let X 1 (H) denote the first column of the complex character table of H. In this article, we show that if X 1 (H) = X 1 (G) , then H ≅ G provided that q + 1 divides neither n nor n – 1. Consequently, it is shown that G is uniquely determined by the structure of its complex group algebra. This in particular extends a recent result of Bessenrodt et al. [Algebra Number Theory 9 (2015), 601–628] to the almost simple groups of arbitrary rank. [ABSTRACT FROM AUTHOR]

For a finite group G, the common divisor graph attached to G is a character degree graph whose vertices are the degrees of the non-linear irreducible complex characters of G, and distinct vertices a, b are adjacent when they have a non-trivial common divisor. In this article, we determine all graphs with five vertices that may occur as for a non-solvable group G. [ABSTRACT FROM AUTHOR]

Let be a finite group and the irreducible character degree set of is contained in , where , and are distinct integers. We show that one of the following statements holds: is solvable; ; or for some prime power . [ABSTRACT FROM AUTHOR]