What's Abelian about abelian groups?

Abstract

The association of names to mathematical concepts and results (the creation of eponyms) is often a curious process. For the case of abelian groups, we will be taken on a quick, guided tour of the life of Niels Henrik Abel, elliptic functions, a curve called the lemniscate, the construction of the regular 17-gon, and a particular class of solvable equations before we can begin to appreciate how Abel's name was attributed to a concept (groups) not yet invented in his lifetime. Therefore, I will have to address how ‘group theory’ was done before it was even invented. As the story unfolds, indications of a broader development in mathematics in the early nineteenth century will emerge. In that century, large parts of analysis underwent transformations from a predominantly formula-centred approach to a more conceptual one, and our story features important examples of how the processes of generalization functioned.

Notes

¹ For more historiographical reflection on such an approach, see, for example, Epple (2011).

² For more analysis of the implicit group-theoretical thought in Gauss' Disquisitiones arithmeticae, see, for example, Neumann (2007), and Wussing (1995).

³ Later, Gauss' analysis of the relation between geometrical construction and quadratic equations would be made rigorous by inter alia Pierre Laurent Wantzel; see also Lützen (2009).

⁴ Abel's life and mathematics has been treated in a number of works. For instance Ore (1957), and Stubhaug (2000) provide good biographies in English. For treatment of Abel's mathematics, see also Houzel (2004) and, in particular, Sørensen (2010a) where more technical details can also be found.

⁵ Translated from: Degen to C Hansteen, 21 May 1821 (Holst et al. 1902, 93).

⁶ The publication of Abel's collected works (Abel 1839) was instrumental in this respect. Among those to comment upon Abel's work and thereby credit him with the results were William Rowan Hamilton and Arthur Cayley.

⁷ For the development of Galois theory and the development of group theory, see, for example, Kiernan (1971), Scholz (1990), and Gray (1990).

⁸ It is remarkable and unfortunate that Toti Rigatelli (1994, 717) got the logic of Abel's reasoning wrong, reproducing the result as ‘he [Abel in Abel (1829)] showed that, in those equations which were solvable by radicals, all roots could be expressed as rational functions of any other root, and that these functions were permutable with respect to the four arithmetical operations. That is, if F ₁ and F ₂ are any two corresponding functional operations, then F ₁ F ₂ x = F ₂ F ₁ x.’

⁹ For a reconstruction of Abel's incomplete investigations in this direction, see Gårding and Skau (1994).

¹⁰ Translated from: Abel to Holmboe, 24 October 1826 (Abel et al., 1902, 44).

¹¹ The decision prompted a brief discussion among Emmy Noether and Helmut Hasse; see Lemmermeyer and Roquette (2006, 175–177, 181).

¹² On this analytical framework of formula-centred and concept-centred mathematics, see also Sørensen (2005, 2009, 2010b).