Prime injections and quasipolarities

TL;DR

This paper characterizes quasipolarities in cyclic groups and provides explicit formulas for their counts based on prime decompositions, also exploring conditions for conjugacy via prime injections.

Contribution

It offers a new explicit formula for counting quasipolarities in cyclic groups and analyzes their conjugacy relations through prime injections.

Findings

Derived explicit formulas for quasipolarity counts.

Established conditions for conjugacy of quasipolarities via injections.

Connected algebraic properties with prime decompositions.

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.