Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Prime injections and quasipolarities   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.

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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
6
The categories of this paper are: astro-ph
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The categories of this paper are: cs
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph