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Die Entdeckung der Sylow-Sätze. (The discovery of the Sylow theorems). (German) Zbl 0637.01006

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.

Reviewer: R.Cooke
##### MSC:
 01A55 Mathematics in the 19th century 20-03 Historical (group theory)
##### Keywords:
Dedekind; abstract group theory; finite groups; p-group