Complete mappings in some linear and projective groups
type: Item
instance of: scholarly article
title: Complete mappings in some linear and projective groups
zbMATH Open document ID: 0783.20013
DOI: 10.1007/BF01207458
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: 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 \(\text{PSL}(2,q)\), \(q\equiv 3\bmod 4\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 446275
zbMATH Keywords: finite linear groups
zbMATH Keywords: finite projective groups
zbMATH Keywords: finite group
zbMATH Keywords: bijective mapping
zbMATH Keywords: complete mapping
zbMATH Keywords: 2-Sylow subgroups
zbMATH Keywords: Mathieu groups
zbMATH Keywords: automorphism groups
author: Francesca Dalla Volta
author: Norberto Gavioli
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 446275