On the admissibility of some linear and projective groups in odd characteristic
Let \(G\) be a finite group and \(\phi\colon G\to G\) be a bijective mapping. Then \(\phi\) is called complete if the mapping \(\eta\), defined by \(\eta(x)=x\phi(x)\), \(x\in G\), is bijective. A group \(G\) is called admissible if it admits a complete mapping. For example, a group of odd order is admissible with the identity map as a complete mapping. M. Hall has shown that a solvable group with non-cyclic Sylow 2-subgroups is admissible. M. Hall and L. J. Paige conjectured that a finite group \(G\) is admissible if and only if a Sylow 2-subgroup of \(G\) is noncyclic or trivial. In the paper under review, the authors answer the conjecture by proving that \(\text{PSL}(2,q)\), \(q\equiv 1\pmod q\), and \(\text{PGL}(2,q)\), \(q\) odd, are admissible. Consequently, they have shown that in odd characteristic the projective groups \(\text{PGL}(n,q)\), \(n\geq 2\) and the linear groups \(\text{GL}(n,q)\), \(n\geq 2\), are also admissible.
scientific article; zbMATH DE number 1053730