The Mathematical Correspondence between Julius Petersen and Ludvig Sylow

The correspondence between the two Scandinavian mathematicians Julius Petersen and Ludvig Sylow is interesting, not only because it sheds light on the researches of the two correspondents, but also because it gives a good impression of the general state of affairs in algebra, more specifically group theory and Galois theory during the early and middle 1870’s, as seen from the nothern fringe of the European mathematical scene. It thus complements Hans Wussing’s vivid and detailed account of this period in his classical The Genesis of the Abstract Group Concept [Wussing 1969/84]. Wussing shows that around 1870 the mathematical avantgarde had emancipated group theory as a separate discipline independent of its use in the theory of equations, thereby paving the way for the introduction of the abstract group concept around 1880. The correspondence between Petersen and Sylow, however, reminds us of the fact that it takes time for such conceptual innovations to filter down through the system. Indeed, for Petersen and Sylow, as well as many of their colleagues, groups were still tied to equations, if they were known at all, and the abstract definition was a far cry of the future. Waterhouse [1980], and in particular Schar-Lau [1988], have emphasized that Sylow’s proof of the famous theorems named after him, built crucially on the fact that he considered groups as Galois groups of a certain equation. In this connection, Scharlau quotes the laconic proof in Petersen’s algebra book [Petersen 1877] of the theorem stating that every group can be considered as a Galois group.