The uniqueness of groups of Lyons type. (English) Zbl 0758.20002
According to the introduction to this paper, the authors ‘give the first computer free proof of the uniqueness of groups of type \(Ly\)’. Previous proofs by C. Sims [Proc. Gainesville Conf. 1972, 138-141 (1973; Zbl 0275.20018)] and W. Meyer, W. Neutsch and R. Parker [Math. Ann. 272, 29-39 (1985; Zbl 0573.20019)] have relied on computer calculations. The present proof relies heavily on another paper by the same authors [Ann. Math., II. Ser. 135, 297-323 (1992)] for definitions and preliminary results, and the two papers need to be read in conjunction. The crucial step is to show that the collinearity graph of the 3-local geometry of any group of type \(Ly\) is simply-connected.
{Note: It is a little unfortunate that in their crusade against computers the authors appear to have lost sight of one advantage that computers do have, namely accuracy. I have not been able to check many of the detailed calculations in this paper, but one or two false assertions stand out. For example, Lemma 2.5(1) is false: \(Z_ 2\times M_{11}\) has not one but two faithful 5-dimensional \(GF(3)\)-representations, which are dual to each other, and only one of them has property (2). In Lemma 5.1(5), \(G_{x,y}\cong S_ 6/Z_ 4\), not \(S_ 6/Z_ 2\)}.
{Note: It is a little unfortunate that in their crusade against computers the authors appear to have lost sight of one advantage that computers do have, namely accuracy. I have not been able to check many of the detailed calculations in this paper, but one or two false assertions stand out. For example, Lemma 2.5(1) is false: \(Z_ 2\times M_{11}\) has not one but two faithful 5-dimensional \(GF(3)\)-representations, which are dual to each other, and only one of them has property (2). In Lemma 5.1(5), \(G_{x,y}\cong S_ 6/Z_ 4\), not \(S_ 6/Z_ 2\)}.
Reviewer: R.Wilson (Birmingham)
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