A note on codegrees and Taketa's inequality

1998

2012

Let G be a nonabelian finite group and let d be an irreducible character degree of G. Then there is a positive integer e so that |G| = d(d+e). Snyder has shown that if e > 1, then |G| is bounded by a function of e. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper, we will show for groups that have a nontrivial, abelian normal subgroup that |G| < e^4 - e^3. We use this to prove that |G| < e^4 + e^3 for all groups. Given that there are a number of solvable groups that meet the first bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.

2022

Berkovich, Chillag and Herzog characterized all finite groups $G$ in which all the nonlinear irreducible characters of $G$ have distinct degrees. In this paper we extend this result showing that a similar characterization holds for all finite solvable groups $G$ that contain a normal subgroup $N$, such that all the irreducible characters of $G$ that do not contain $N$ in their kernel have distinct degrees.

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.

Bulletin of the Australian Mathematical Society, 1977

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) -X(u) = c or X(u) = 0 . It is shown that c is an even positive integer and all such groups with a -8 are described.

Monatshefte für Mathematik, 2015

For a finite non-abelian group G let rat(G) denote the largest ratio of degrees of two nonlinear irreducible characters of G. We prove that the number of non-abelian composition factors of G is bounded above by 1.8 ln(rat(G)) + 1.3.

Israel Journal of Mathematics, 2013

Journal of Algebra, 2004

A longstanding problem in the representation theory of finite solvable groups, sometimes called the Taketa problem, is to find strong bounds for the derived length dl(G) in terms of the number |cd(G)| of irreducible character degrees of the group G. For p-groups an old result of Taketa implies that dl(G) ≤ |cd(G)|, and while it is conjectured that the true bound is much smaller (in fact, logarithmic) for large dl(G), it turns out to be extremely difficult to improve on Taketa's bound at all. Here, therefore, we suggest to first study the problem for a restricted class of p-groups, namely normally monomial p-groups of maximal class. We exhibit some structural features of these groups and show that if G is such a group, then dl(G) ≤ 1 2 |cd(G)| + 11 2 .

Proceedings of the American Mathematical Society, 1999

Let b(G) denote the largest irreducible character degree of a finite group G, and let p be a prime. Two results are obtained. First, we show that, if G is a p-solvable group and if b(G) < p 2 , then p 2 | G : Op(G)|. Next, we restrict attention to solvable groups and show that, if b(G) ≤ p α and if P is a Sylow p-subgroup of G, then |P : Op(G)| ≤ p 2α .

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.