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

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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.

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