Prime injections and quasipolarities

Abstract

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$e^{w}.r:=(w,r)\in \mathbb{Z}/pn \mathbb{Z}\rtimes \mathbb{Z}/pn\mathbb{Z}^{\times}$. Suppose that $e^{u}.v\in\mathbb{Z}/n\mathbb{Z}\rtimes \mathbb{Z}/n\mathbb{Z}^{\times}$ is seen as an automorphism of $\mathbb{Z}/n\mathbb{Z}$defined by $e^{u}.v(x)=vx+u$; then $e^{u}.v$ is a \emph{quasipolarity} if it is an involution without fixed points.In this brief note we give an explicit formula for the number of quasipolarities of $\mathbb{Z}/n\mathbb{Z}$ interms 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.