A computer-free construction of the third group of Janko
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 \(Z(\text{SU}_9(2))\).
\(9\)-dimensional representations over \(\text{GF}(4)\)
scientific article; zbMATH DE number 1038267
scientific article