Complete mappings in some linear and projective groups
Let \(G\) be a finite group; a bijective mapping \(\varphi: G\to G\) is complete if also the mapping \(x\mapsto x\varphi(x)\) is bijective. \textit{M. Hall} and \textit{L. J. Paige} [Pac. J. Math. 5, 541-549 (1955; Zbl 0066.27703)] proved that a necessary condition for a finite group to have a complete mapping is that its 2-Sylow subgroups be non-cyclic or trivial. They conjectured such that a condition is also sufficient and proved that this is true for solvable, symmetric and alternating groups. In our paper it is proved that, in even characteristic, the groups \(\text{SL}(n,q)\), \(\text{PSL}(n,q)\), \(\text{GL}(n,q)\) and \(\text{PGL}(n,q)\), admit a complete mapping, with the exception of \(\text{SL}(2,2)\). Moreover, the same is proved for the Mathieu groups, their automorphism groups and \(\text{PSL}(2,q)\), \(q\equiv 3\bmod 4\).
scientific article; zbMATH DE number 446275