On $ {p} $-groups of maximal class

• Published in: Mathematische Nachrichten 2011-03, Vol.284 (4), p.471-493
• Main Authors: Jaikin-Zapirain, A., Vera-López, A.
• Format: Article
• Language: English
• Subjects: 20D15; Finite groups; maximal class; MSC
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.200710198

Let R be the ring \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ {\mathbb Z}[x]/\left({{x^p-1}\over{x-1}}\right) = {\mathbb Z}[\bar{x}] $\end{document} and let \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathfrak {a} $\end{document} be the ideal of R generated by $ (\bar{x}-1) $. In this paper, we discuss the structure of the \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ {\mathbb Z}[C_p] $\end{document}‐module \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ (R/\mathfrak {a}^{n-1}) \wedge (R/\mathfrak {a}^{n-1}) $\end{document}, which plays an important role in the theory of p‐groups of maximal class (see 2–5). The generators of this module allow us to obtain the defining relations of some important examples of p‐groups of maximal class with Y1 of class two. In particular we obtain the best possible estimates for the degree of commutativity of p‐groups of maximal class with Y1 of class two. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim