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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: A. Jaikin-Zapirain and Alexander Moretó
Journal: Trans. Amer. Math. Soc. 354 (2002), 3907-3925
MSC (2000): Primary 20C15; Secondary 20E18
Published electronically: April 12, 2002
MathSciNet review: 1926859
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Abstract: Let $\mathcal{S}$ be a finite set of powers of $p$ containing 1. It is known that for some choices of $\mathcal{S}$, if $P$ is a finite $p$-group whose set of character degrees is $\mathcal{S}$, then the nilpotence class of $P$ is bounded by some integer that depends on $\mathcal{S}$, while for some other choices of $\mathcal{S}$ 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 $\mathcal{S}$ is class bounding if and only if $p\notin\mathcal{S}$. 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 $\mathcal{S}$ such that $p\notin\mathcal{S}$.

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Additional Information

A. Jaikin-Zapirain
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco Ciudad Universitaria, 28049 Madrid, Spain

Alexander Moretó
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain

Received by editor(s): July 18, 2001
Received by editor(s) in revised form: December 17, 2001
Published electronically: April 12, 2002
Additional Notes: Research of the first author partially supported by DGICYT. Research of the second author supported by the Basque Government and the University of the Basque Country.
Article copyright: © Copyright 2002 American Mathematical Society