On Distinct Character Degrees

2016

Let G be a finite group. We say that the derived covering number of G is finite if and only if there exists a positive integer n such that C n = G ′ for all non-central conjugacy classes C of G. In this paper we characterize solvable groups G in which the derived covering number is finite.

Let G be a finite group‎. ‎We say that the derived covering number of G is finite if and only if there exists a positive integer n such that Cn=G′ for all non-central conjugacy classes C of G‎. ‎In this paper we characterize solvable groups G in which the derived covering number is finite‎.‎

Israel Journal of Mathematics, 1991

Let G be a finite group. The co-degree of an irreducible character X of G is defined to be the number I GI/X (1). The set of all prime divisors of all the co-degrees of the nonlinear irreducible characters of G is denoted by E(G). First we show that (G) = lr (G) (the set of all prime divisors of I G]) unless G is nilpotent-by-abelian. Then we make I;(G) a graph by adjoining two elements of E(G) if and only if their product divides a co-degree of some nonlinear character of G. We show that the graph E(G) is connected and has diameter at most 2. Additional information on the graph is given. These results are analogs to theorems obtained for the graph corresponding to the character degrees (by Manz, Staszewski, Willems and Wolf) and for the graph corresponding to the class sizes (by Bertram, Herzog and Mann). Finally, we investigate groups with some restriction on the co-degrees. Among other results we show that if G has a co-degree which is a p-power for some prime p, then the corresponding character is monomial and Op(G) #: 1. Also we describe groups in which each co-degree of a nonlinear character is divisible by at most two primes. These results generalize results of Chillag and Herzog. Other resuits are proved as well.

Archiv der Mathematik, 2013

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then dl(G) |cd(G)|, where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality dl(N) |cd(G | N)| holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if cd(G | N) is a set of non-trivial p-powers for some fixed prime p.

Journal of Algebra, 2005

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.

2002

Journal of Algebra, 2001

Journal of Algebra, 2001

Manuscripta Mathematica, 2015

Let G be a finite group and d the degree of a complex irreducible character of G, then write |G| = d(d + e) where e is a nonnegative integer. We prove that |G| ≤ e 4e 3 whenever e > 1. This bound is best possible and improves on several earlier related results.

Proceedings of the American Mathematical Society, 1997

Given a finite solvable group G, we say that G has property P k if every set of k distinct irreducible character degrees of G is (setwise) relatively prime. Let k(G) be the smallest positive integer such that G satisfies property P k. We derive a bound, which is quadratic in k(G), for the total number of irreducible character degrees of G. Three exceptional cases occur; examples are constructed which verify the sharpness of the bound in each of these special cases.