# Automorphisms of a Clifford-like parallelism

## TL;DR

This paper investigates the automorphism group of a Clifford-like parallelism on a 3D projective space over a quaternion skew field, comparing it with the automorphism group of the left parallelism and exploring various examples.

## Contribution

It characterizes the automorphism group of Clifford-like parallelisms and compares it with the automorphism group of the left parallelism, revealing conditions for their equality or difference.

## Key findings

- Automorphism group can be equal to or properly contained in the left parallelism automorphism group.
- Examples show that different quaternion skew fields can lead to different automorphism group structures.
- The structure of automorphism groups depends on the choice of the Clifford-like parallelism.

## Abstract

In this paper we focus on the description of the automorphism group $\Gamma_{\parallel}$ of a Clifford-like parallelism $\parallel$ on a $3$-dimensional projective double space $\bigl(\mathbb{P}(H_F),{\mathrel{\parallel_{\ell}}},{\mathrel{\parallel_{r}}}\bigr)$ over a quaternion skew field $H$ (with centre a field $F$ of any characteristic). We compare $\Gamma_{\parallel}$ with the automorphism group $\Gamma_{\ell}$ of the left parallelism $\mathrel{\parallel_{\ell}}$, which is strictly related to $\mathrm{Aut}(H)$. We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose $\parallel$ in such a way that $\Gamma_{\parallel}$ is either properly contained in $\Gamma_{\ell}$ or coincides with $\Gamma_{\ell}$ even though ${\parallel} \neq{\mathrel{\parallel_{\ell}}}$.