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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]

LetHbe a finite group,n≥3, andqbe a prime power such thatq−1 divides neithernnorn−1. Denote byX1(H) the first column of the ordinary character table ofH. LetGbe a finite group such that. In this paper, we will show that if, thenH≅G. As a consequence, we show that the almost simple groupsGare uniquely determined by the structure of their complex group algebras. This extends a result of Tong-Viet [30] to a family of almost simple groups of Lie type of arbitrary rank. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Let X ( G) be the first column of the ordinary character table of G. We will show that if X ( G) = X1(PGU( q )), then G ≅ PGU( q ). As a consequence, we show that the projective general unitary groups PGU( q ) are uniquely determined by the structure of their complex group algebras. [ABSTRACT FROM AUTHOR]