The discovery of the Sylow theorems
type: Item
instance of: scholarly article
title: The discovery of the Sylow theorems
zbMATH Open document ID: 0637.01006
DOI: 10.1016/0315-0860(88)90048-1
author: Winfried Scharlau
published in: Historia Mathematica
publication date: 1988-01-01T00:00:00Z
review text: The author notes the discovery of Dieudonné that Galois had asserted the essence of the Sylow theorems in a marginal note. According to the author, it was Dedekind who took up the challenge in Galois' legacy. The basis of the author's thesis is that there were no essential barricades to the development of an abstract group theory after 1850, that Cauchy's theorem pointed in the direction of a classification of finite groups which would necessarily lead to the Sylow theorems; and that Dedekind, knowing that a \(p\)-group has a non-trivial center, could have deduced the solvability of \(p\)-groups directly. The author traces the route followed by Sylow from his studies, while a student of Bjerknes, of Abel's incomplete works on algebraic equations, through his first publication (in 1867) of a theorem intermediate between that of Galois and the full Sylow theorems, to a letter of 19 September 1870, in which he indicates that he is in possession of the full theorem, though he applies it only for \(p=2\) and gives no proof. Sylow's statement of the theorem is in terms of Galois groups rather than abstract groups, for reasons the author makes clear in the subsequent section.
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 20-03
zbMATH DE Number: 4037797
zbMATH Keywords: Dedekind
zbMATH Keywords: abstract group theory
zbMATH Keywords: finite groups
zbMATH Keywords: p-group
Wikidata QID: Q56092195
reviewed by: Roger L. Cooke
MaRDI profile type: Publication
full work available at URL: 0315-0860(88)90048-1
OpenAlex ID: W2054096149
cites work: Q5628053
cites work: Q3221913
cites work: Q5586187
cites work: Endliche Gruppen I
cites work: The early proofs of Sylow's theorem
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description: scientific article; zbMATH DE number 4037797