On the existence and uniqueness of the sporadic simple groups \(J_2\) and \(J_3\) of Z. Janko (Q2711634)

There exist quite a many existence and/or uniqueness proofs for \(J_2\) and \(J_3\). For \(J_2\) see the classical proof due to \textit{M. Hall} jun. and \textit{D. Wales} [J. Algebra 9, 417-450 (1968; Zbl 0172.03103)] and the one due to \textit{R. Weiss} [Math. Proc. Camb. Philos. Soc. 108, No.~1, 7-19 (1990; Zbl 0709.51009)]. For \(J_3\) we have the classical one due to \textit{G. Higman} and \textit{J. McKay} [Bull. Lond. Math. Soc. 1, 89-94 and 219 (1969; Zbl 0175.30103)], the papers due to \textit{M. Aschbacher} [Geom. Dedicata 35, No.~1-3, 143-154 (1990; Zbl 0703.20013)] embedding \(J_3\) into \(E_6(4)\), \textit{D. Frohardt} [J. Algebra 83, 349-379 (1983; Zbl 0518.20012)] as an automorphism group of a trilinear form, \textit{R. Weiss} [Math. Z. 179, 91-95 (1982; Zbl 0481.05034), Trans. Am. Math. Soc. 298, 621-633 (1986; Zbl 0606.20021)], acting on a certain graph and quite recently due to \textit{B. Baumeister} [J. Algebra 192, No.~2, 780-809 (1997; Zbl 0879.20008)] embedding into \(\text{SU}_9(4)\).NEWLINENEWLINENEWLINEIn the paper under review the author starts with the fact that both groups are completions of a rank 2 amalgam \(H,M\), where \(H\) is the centralizer of an involution in \(J_2\) or \(J_3\) and \(M\) is the normalizer of a fours-group. Now let \(S\in\text{Syl}_2(H)\), and \(T\) be a Sylow 3-subgroup of \(N_G(S)\). Then \(N_G(T)/T\cong\text{PGL}_2(9)\) in both groups. But in \(J_2\) we have a nonsplit extension while in \(J_3\) the extension splits. The author now investigates the amalgam of the three groups \(H\), \(M\), \(N_r\), \(r=2,3\); \(N_2\) the nonsplit group and \(N_3\) the split one. He proves that the universal covering in both cases is finite. In the first case it is of order \(2^7\cdot 3^3\cdot 5^2\cdot 7\) while in the second it is of order \(2^7\cdot 3^6\cdot 5\cdot 17 \cdot 19\). This has been done by establishing generators and relations for the amalgams and then using coset enumeration. It is a remarkable fact that the relations differ just at one place. If we call a simple group \(G\) of \(J_r\) type (\(r=2,3\)) if there is an involution \(z\) with \(C_G(z)\cong 2^{1+4}A_5\) and \(G\) has two classes of involutions for \(r=2\) and one for \(r=3\), then using the results of \textit{Z. Janko} [Symp. Math. 1, 25-64 (1969; Zbl 0182.35304)], this shows existence and uniqueness of \(J_2\) and \(J_3\).