A computer-free construction of the third group of Janko
type: Item
instance of: scholarly article
title: A computer-free construction of the third group of Janko
zbMATH Open document ID: 0879.20008
author: Barbara Baumeister
published in: Journal of Algebra
publication date: 1998-01-25T00:00:00Z
review text: The author gives a new and elementary existence proof for \(J_3\). This group discovered by Z. Janko in 1968 was shown to exist by \textit{G. Higman} and \textit{J. McKay} [Bull. Lond. Math. Soc. 1, 89-94 (1969; Zbl 0175.30103)] with the aid of a computer. The first computer free existence proof was given by \textit{R. Weiss} [Math. Z. 179, 91-95 (1982; Zbl 0481.05034)] using a complicated graph of diameter 10. Later \textit{M. Aschbacher} gave a computerfree existence proof [Geom. Dedicata 35, No. 1-3, 143-154 (1990; Zbl 0703.20013)] using an embedding in \(E_6(4)\) and so a 27-dimensional representation over \(\text{GF}(4)\). The existence proof given by the author uses an embedding in \(U_9(2)\) and so a 9-dimensional representation over \(\text{GF}(4)\), the smallest one. The starting point is an amalgam of three groups \(G_1=L_2(16)2\), \(G_2=E_{16}(Z_3\times A_5)\) and \(G_3=3\text{PGL}_2(9)\). This amalgam gives rise to a so called \(c_2c\)-geometry. From the theory of these geometries one gets an upper bound on the number of points and so an upper bound on the order of any group \(G\) possessing such an amalgam. Nevertheless the universal completion might be the trivial group. The next step is to embed the amalgam into \(U_9(2)\), which proves that the universal completion is nontrivial. Now using the upper bound on the order of \(G=\langle G_1,G_2,G_3\rangle\leq U_9(2)\) the author is able to show that \(G\) is simple, all involutions are conjugate and for \(i\) an involution \(C_G(i)\) is an extension of an extraspecial group of order 32 by \(A_5\). This is the definition of \(J_3\), and so it is proven that \(J_3\) exists. As a corollary one gets the existence of \(3\times J_3\), using the element of order 3 in \(Z(\text{SU}_9(2))\).
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20C34
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 1038267
zbMATH Keywords: existence proof for \(J_ 3\)
zbMATH Keywords: embeddings in \(U_ 9(2)\)
zbMATH Keywords: \(9\)-dimensional representations over \(\text{GF}(4)\)
zbMATH Keywords: \(c_ 2c\)-geometries
zbMATH Keywords: numbers of points
zbMATH Keywords: amalgams
zbMATH Keywords: universal completions
zbMATH Keywords: involutions
reviewed by: Gernot Stroth
MaRDI profile type: Publication
full work available at URL: jabr.1996.6954
OpenAlex ID: W2047687810
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DOI: 10.1006/JABR.1996.6954
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description: scientific article; zbMATH DE number 1038267