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 $e^{w}.r:=(w,r)\in \mathbb{Z}_{p n}\rtimes \mathbb{Z}_{p n}^{\times}$. 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.