Prime injections and quasipolarities

• Published in: arXiv.org 2013-10
• Main Author: Agustín-Aquino, Octavio Alberto
• Format: Article
• Language: English
• Subjects: Automorphisms
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1302.6874

Let \(p\) be a prime number. Consider the injection \[ \iota:\mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/pn\mathbb{Z}:x\mapsto px, \] and the elements \(e^{u}.v:=(u,v)\in \mathbb{Z}/n\mathbb{Z}\rtimes \mathbb{Z}/n\mathbb{Z}^{\times}\) and \(e^{w}.r:=(w,r)\in \mathbb{Z}_{p n}\rtimes \mathbb{Z}_{p n}^{\times}\). Suppose \(e^{u}.v\in \mathbb{Z}/n\mathbb{Z}\rtimes \mathbb{Z}/n\mathbb{Z}^{\times}\) is seen as an automorphism of \(\mathbb{Z}/n\mathbb{Z}\) by \(e^{u}.v(x)=vx+u\); then \(e^{u}.v\) is a quasipolarity if it is an involution without fixed points. In this brief note give an explicit formula for the number of quasipolarites of \(\mathbb{Z}/n\mathbb{Z}\) in terms of the prime decomposition of \(n\), and we prove sufficient conditions such that \((e^{w}.r)\circ \iota =\iota\circ (e^{u}.v)\), where \(e^{w}.r\) and \(e^{u}.v\) are quasipolarities.