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

Abstract

Abstract. Let S be a finite set of powers of p containing 1. It is known that for some choices of S, if P is a finite p-group whose set of character degrees is S, then the nilpotence class of P is bounded by some integer that depends on S, while for some other choices of S such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set S is class bounding if and only if p / ∈ S. In this article we provide a new approach to this problem. Our main result shows the relevance of certain p-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets S such that p / ∈ S.

Characters of p-groups and Sylow p-subgroups

“…All the class bounding sets S found in that paper have the property that p ∈ S. In [19] Isaacs and Slattery prove that this a necessary condition for a set S to be class bounding. However, an example constructed in [20] shows that this is not a sufficient condition. In [20] Jaikin-Zapirain and the author also find more class bounding sets.…”

“…However, an example constructed in [20] shows that this is not a sufficient condition. In [20] Jaikin-Zapirain and the author also find more class bounding sets. We refer the reader to [16,19] and [20] for the detailed results and to [20] for some specific questions related to this problem.…”

“…It is well-known that this number has a strong influence on the structure of the group P (see, for instance, Problem 5.14 of [11] and [26]). In particular, we want to stress the influence of m(P ) in the last problem discussed in the previous section (see [20]). We introduce a new invariant associated to any p-group.…”

GVZ-groups, Flat groups, and CM-Groups

“…In fact, there has been some research on which sets of character degrees bound the nilpotence class of a p-group. (See [11,12] and [13]. )…”

Partial GVZ-groups

“…In fact, there has been some research on which sets of character degrees bound the nilpotence class of a p-group. (See [12], [13], and [14]. )…”