On the admissibility of some linear and projective groups in odd characteristic
type: Item
instance of: scholarly article
title: On the admissibility of some linear and projective groups in odd characteristic
zbMATH Open document ID: 0896.20017
DOI: 10.1023/A:1004924402272
author: Francesca Dalla Volta
author: Norberto Gavioli
published in: Geometriae Dedicata
publication date: 1998-10-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 1053730
zbMATH Keywords: admissible groups
zbMATH Keywords: finite groups
zbMATH Keywords: complete mappings
zbMATH Keywords: solvable groups
zbMATH Keywords: Sylow 2-subgroups
zbMATH Keywords: projective groups
zbMATH Keywords: linear groups
reviewed by: Qaiser Mushtaq
MaRDI profile type: Publication
full work available at URL: a:1004924402272
OpenAlex ID: W66896227
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description: scientific article; zbMATH DE number 1053730