Complete mappings stabilizing Sylow $2$-subgroups in central quotients of special linear groups

Abstract

A complete mapping of a finite group $G$ is a permutation $\theta$ of $G$ such that $x\mapsto x\theta (x)$ is also a permutation of $G$. Complete mappings have applications in cryptography, particularly in the design of round functions for cryptographic systems. The study of their parity has led to the question of which finite groups admit complete mappings of both parities. A useful approach to this question is to construct complete mappings stabilizing suitable subgroups, in particular Sylow $2$-subgroups. We show that, for every $n\ge 2$, every prime power $q$, and every $Z\le Z({\mathrm{SL}}_n(q))$, the group ${\mathrm{SL}}_n(q)/Z$ admits a complete mapping stabilizing a Sylow $2$-subgroup if and only if $(n,q)\ne (2,2)$.