Received January 17, 2026. Published online July 17, 2026.
Abstract: An element of a group is called reversible if it is conjugate to its inverse. While reversibility in the quaternionic Möbius group ${\rm PSL}(2,\mathbb{H})$ has traditionally been studied using geometric and dynamical methods, we develop a purely algebraic approach. We obtain an explicit, computable criterion for the reversibility of a quaternionic Möbius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that $[A]\in{\rm PSL}(2,\mathbb{H}) \text{is reversible} \Longleftrightarrow\beta_A^2=\delta_A^2$, where $\beta_A$ and $\delta_A$ are real conjugacy invariants associated with a lift $A\in{\rm SL}(2,\mathbb{H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in ${\rm SL}(2,\mathbb{H})$ and ${\rm PSL}(2,\mathbb{H})$.
Affiliations: Krishnendu Gongopadhyay, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar 140306, Punjab, India, e-mail: krishnendu@iisermohali.ac.in; Tejbir Lohan (corresponding author), Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi Centre, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India, e-mai: tejbirlohan70@gmail.com; Abhishek Mukherjee, Kalna College, Kalna, Dist. Burdwan, P.O. Kalna, Bardhaman, West Bengal 713409, India, e-mail: abhishekmukherjee@kalnacollege.ac.in
