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In this paper, we determine all almost simple groups each of whose character degrees has at most two distinct prime divisors. More generally, we show that a finite non-solvable group [Formula: see text] with this property is an extension of an almost simple group [Formula: see text] by a solvable group and [Formula: see text], where [Formula: see text] is the set of all primes dividing some character degree of [Formula: see text].

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A digraph − → PC(G) is said to be the directed power graph on the conjugacy classes of a group G, if its vertices are the non-trivial conjugacy classes of G, and there is an arc from vertex C to C ′ if and only if C � = C ′ and C C ′m for some positive integer m > 0. Moreover, the simple graph PC(G) is said to be the (undirected) power graph on the conjugacy classes of a group G if its vertices are the conjugacy classes of G and two distinct vertices C and C ′ are adjacent in PC(G) if one is a subset of a power of the other. In this paper, we find some connections between algebraic properties of some groups and properties of the associated graph.

Let G be a finite group. We define the derived covering number and the derived character covering number of G, denoted respectively by dcn (G) and dccn (G), as the smallest positive integer n such that Cn = G′ for all non-central conjugacy classes C of G and Irr ((χn)G′) = Irr (G′) for all nonlinear irreducible characters χ of G, respectively. In this paper, we obtain some results on dcn and dccn for a finite group G, such as the existence of these numbers and upper bounds on them.