For a character χ of a finite group G, the number a(χ) := |G : ker χ |/χ (1) is called the co-degree of χ . The object of this paper is to study the connection between the structure of a finite group and the co-degrees of its irreducible characters.

The result of this note is as follows. If a finite solvable group has an element of order m, then the group has an irreducible character whose codegree contains all prime divisors of m.Mathematics Subject Classification (2010). Primary 20C20, 20C15.

For an irreducible character χ of a finite group G, the codegree of χ is defined by |G:kerχ|/χ(1). In this note, we show that if a finite solvable group G has an element of order m, then G admits an irreducible character of codegree divisible by m.

No abstract

D e d i c a t e d t o P r o f . O t t o K e g e l o n h i s 7 0 t h b i r t h d a yAbstract. In this note we classify the finite groups satisfying the following property P 5 : their conjugacy class lengths are set-wise relatively prime for any 5 distinct classes.

No abstract

In this paper non-nilpotent groups with two irreducible character degrees are characterized. This is done using a description of solvable groups in which the commutator subgroup is a minimal normal subgroup.

Let G be a finite group and p be a prime. Assume that G has no irreducible character of degree divisible by p2. Then false|G/Opfalse(Gfalse)|p⩽p3.