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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]