Prime injections and quasipolarities

Abstract

Let $p$ be a prime number. Consider the injection \[ ι:\mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/pn\mathbb{Z}:x\mapsto px, \] and the elements $e^u.v:=(u,v)\in \mathbb{Z}/n\mathbb{Z}⋊\mathbb{Z}/n{\mathbb{Z}}^{\times }$ and $e^w.r:=(w,r)\in {\mathbb{Z}}_{pn}⋊{\mathbb{Z}}_{pn}^{\times }$. Suppose $e^u.v\in \mathbb{Z}/n\mathbb{Z}⋊\mathbb{Z}/n{\mathbb{Z}}^{\times }$ 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.