On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan have Proved the Jordan-Hölder Theorem?

Abstract

In 1870 Jordan proved that the composition factors of two composition series of a group are the same. Almost 20 years later Holder (1889) was able to extend this result by showing that the factor groups, which are quotient groups corresponding to the composition factors, are isomorphic. This result, nowadays called the Jordan-Holder Theorem, is one of the fundamental theorems in the theory of groups. The fact that Jordan, who was working in the framework of substitution groups, was able to prove only a part of this theorem is often used to emphasize the importance and even the necessity of the abstract conception of groups, which was employed by Holder. However, as a little-known paper from 1873 reveals, Jordan had all the necessary ingredients to prove the Jordan-Holder Theorem at his disposal (namely, composition series, quotient groups, and isomorphisms), and he also noted a connection between composition factors and corresponding quotient groups. Thus, I argue that the answer to the question posed in the title is “Yes.” It was not the lack of the abstract notion of groups which prevented Jordan from proving the Jordan-Holder Theorem, but the fact that he did not ask the right research question that would have led him to this result. In addition, I suggest some reasons why this has been overlooked in the historiography of algebra, and I argue that, by hiding computational and cognitive complexities, abstraction has important pragmatic advantages.