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

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Abstract

In 1870 Jordan proved that the composition factors of two composition series of a group are the same. Almost 20 years later Hölder (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-Hölder 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 Hölder. However, as a little-known paper from 1873 reveals, Jordan had all the necessary ingredients to prove the Jordan-Hölder 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-Hölder 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.

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Notes

  1. “In mathematics the art of asking questions is to be practiced more often than that of solving problems.” Cited from Fraenkel (1932, p. 454).

  2. A substitution is an operation that changes a given arrangement of things into a different arrangement (permutation) of the same things. Historically, the terms “permutation group” and “substitution group” have often been used interchangeably. See, e.g., the title of Cauchy (1844) and also Miller (1935, p. 8). I shall employ the more frequently used term “substitution group.”

  3. Notable exceptions are Cayley (1854) and Dedekind (1855–58). On Dedekind, see also Scharlau (1981).

  4. This sentiment is expressed, for example, by Scholz (1990, p. 396) and Corry (1996, pp. 24–36).

  5. This is suggested, for example, by Wussing (1984, p. 244), Scholz (1990, p. 396), and Nicholson (1993, p. 84).

  6. N is a maximal normal subgroup of G, if no other normal subgroup of G has a greater order than N. In general, a group can have different maximal normal subgroups.

  7. The order of a group G, i.e., its number of elements, is denoted by |G|.

  8. For example, the quaternion group of order eight has three, the dihedral group of order eight has seven different composition series (Dummit and Foote 1991, p. 106).

  9. Hölder calls the elements of a group “operations,” and uses H instead of N.

  10. See Nicholson (1993), but also Wussing (1984) and van der Waerden (1985).

  11. These quotient groups are nowadays somewhat misleadingly called composition factors. See, e.g., (Dummit and Foote 1991, p. 103). Nicholson also deplores this terminology (Nicholson 1993, p. 83). I will retain Hölder’s terminology in this paper.

  12. See also Sylow (1872, p. 585), Netto (1878), and Capelli (1878, pp. 43–45).

  13. The idea of using an equivalence relation instead of identity goes back to Gauss’s work on congruence classes (1801).

  14. Thus, s and t are congruent modulo N, if they lie in the same coset: sN = tN amounts to s · h 1 = t · h 2, i.e., s = t · (h 2 · h −11 ), for some h 1, h 2 ∈ N. Since both h 2 and h −11 are in the subgroup N, h 2 · h −11 is also an element, say h, of N. Jordan uses H instead of N.

  15. In Jordan (1873b), according to Nicholson (1993, p. 75).

  16. A subgroup that commutes with all elements of the group is a normal subgroup.

  17. Neither Hölder (1889) nor Dyck (1880), who is the only author referred by Hölder in connection with quotient groups, refer to Jordan’s 1873 paper. Both refer only to Jordan’s treatment of composition factors in (Jordan 1870). Hölder mistakenly writes: “The [factor] groups do not occur in the works of Jordan and Netto, where the factors of a composition are understood as pure numbers” (Hölder 1889, p. 34; original emphasis).

  18. A very similar proof is also presented in a little-known paper by Brown (1895) and in (Burnside 1911, pp. 65–68).

  19. For the setup of the involved groups, see the sketch of Hölder’s proof above.

  20. Nicholson ignores this difference, claiming instead that Weber “went on to prove the Jordan-Hölder Theorem in the same way that Hölder had proved it in 1889, using quotient groups” (Nicholson 1993, p. 83). Indeed, Weber himself claims, somewhat misleadingly, that he is following Hölder’s proof (Weber 1896, p. 18), who in turn describes his proof as a modification of Netto’s train of thoughts (Hölder 1889, p. 34). Brown’s assessment is closer to the mark: His presentation is also built up on Netto’s, but he describes his proof as a “very much simplified” version compared to that of Hölder (Brown 1895, p. 232).

  21. See, e.g., (Frobenius and Stickelberger 1879, p. 222): They distinguish between the equality of the group elements and the “relative equality” of the elements of a quotient, and then remark that they present the proofs of a number of theorems only for the former notion in order “to make the presentation more convenient,” but that they carry over also to the latter.

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Acknowledgements

This paper was presented at Meeting of the Canadian Society for History and Philosophy of Mathematics in Toronto (May 2006), at the GAP.6 Workshop Towards a New Epistemology of Mathematics in Berlin, (September 2006), and at the Meeting of the Canadian Mathematical Society in Toronto (December 2006). The author would like to thank the audiences, in particular Leo Corry, for their valuable comments. Translations are by the author, unless noted.

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    Dirk Schlimm

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Schlimm, D. On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan have Proved the Jordan-Hölder Theorem?. Erkenn 68, 409–420 (2008). https://doi.org/10.1007/s10670-008-9108-z

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