Orbit sizes and character degrees

Abstract

The objective of this paper is to develop criteria that guarantee that a finite group G which acts faithfully on a vector space V possesses "many" orbits of different sizes. This has consequences on (classical and modular) character degrees of finite solvable groups.Introduction.

Character degrees and nilpotence class of finite 𝑝-groups: An approach via pro-𝑝 groups

“…We also write d(G) to denote the minimal number of generators of G. We recall that the rank of a finite group G (we denote it by rk(G)) is the maximum of the numbers d(H), as H runs over the subgroups of G. We have been unable to find any reference to results of this type when the field is infinite. T. M. Keller has obtained results of this flavor for finite fields (see, for instance, Theorem 2.1 of [12]). We do not know whether a linear bound for the number of generators (or, perhaps, even for the rank) exists.…”

Derived length and character degrees of solvable groups

“…However, it is believed that the "right" bound is logarithmic and a lot of work has recently been done in this direction by T. Keller, who has essentially reduced the problem to p-groups. (See [13], [14], [15].) We believe that it should also be possible to improve the bound in Theorem A to a logarithmic bound, but for that, one would need to solve first the p-group case of the original problem.…”

Fitting Heights of Solvable Groups with Few Character Degrees

“…(ii) In the notation of Lemma 2.10, the group E × A of order 2 2 · 3 · 5 has a normal subgroup of order 5 and acts faithfully on the elementary abelian group E of order 2 4 . Therefore E × A is isomorphic to a subgroup of N. But (i) tells us that E × A = N .…”