An exposition of Jordan's original proof of his theorem on finite subgroups of \(\mathrm{GL}_n (\mathbb{C})\) (Q6628975)

In the paper under review the author discusses a famous result by \textit{C. Jordan} [Borchardt J. LXXXIV, 89--215 (1877; JFM 09.0096.01)] which states that if \(G\) is a finite subgroup of \(\mathrm{GL}_{n}(\mathbb{C})\), then there is a normal abelian subgroup \(A\trianglelefteq G\) such that \(|G:A| \leq J(n)\), where \(J(n)\) depends only on \(n\). Jordan argued by induction on the dimension, but he gave no explicit bound on \(J(n)\) in his article, not even an inductive one.\N\NJordan's original proof was based on a purely algebraic idea. He enumerates the elements of \(G\) according to the shape and size of their centralizers and one can thus write a class equation involving the order of \(G\) and of the centralizers of its elements. Inducting on dimension, this yields a Diophantine equation of the form\N\[\N\frac{1}{g}=\frac{1}{q_{1}} + \dots + \frac{1}{q_{k}} -\frac{a}{b}\N\]\Nwhere \(g=|G: Z(G)|\), \(a\), \(b\) and \(k\) are integers that are bounded in terms of \(n\) only and each \(q_{i}\) is the cardinality of a certain subgroup of \(G/Z(G)\). Reasoning by induction the author proves that \(J(n) \leq (k!a)^{2^{k}}\) (see Lemma 4.1).\N\NThe author mentions in passing the proof of the Landau theorem [\textit{E. Landau}, Math. Ann. 56, 671--676 (1903; JFM 34.0241.09)] that there are only finitely many finite groups \(G\) with exactly \(k\) conjugacy classes \(\operatorname{cl}_{1}, \dots \operatorname{cl}_{k}\) is based on a similar, and easier, Diophantine equation, namely \(\frac{1}{q_{1}}+\dots +\frac{1}{q_{k}}=1\) where \(q_{i}=|G|/|\operatorname{cl}_{i}|\). The reviewer points out that (using the greedy algorithm to express 1 as an Egyptian fraction) it is possible to prove that \(q_{i} \leq s(i)\) where \(s(1)=2\) and \(s(n+1)=s(n)^{2}-s(n)+1\) if \(n>1\). So \(k\geq \log_{2}\log_{2}(|G|)\).\N\NFinally the author asserts that: ``From a purely epistemological viewpoint, it is however interesting to consider how Jordan gets away with not writing down any bound whatsoever in his original memoir. In fact, in order to convince the reader of the soundness of his argument, he introduces a distinction between two kinds of numbers, which he calls limited and unlimited. [...] A century and a half after Jordan, it is hard not to see there the premise of a way of thinking that prefigures nonstandard analysis, where a new kind of number, the unlimited ones, is given an existence of its own.''\N\NThe reviewer does not want to discuss this delicate issue. He recalls the acrimonious controversy initiated by \textit{S. Unguru} [Arch. Hist. Exact Sci. 15, 67--114 (1975; Zbl 0325.01002)] (see also [Isis 70, 555--564 (1979; Zbl 0424.01003)]) against ``The approach [to the history of mathematics] of Freudenthal, van der Waerden, and their cohort'' (see a reply by \textit{A. Weil} [Arch. Hist. Exact Sci. 19, 91--93 (1978; Zbl 0393.01001)]).\N\NIn any case, the author takes his previous declaration as a starting point to give a non-standard treatment of Jordan's proof.