A semiclassical parabolic system related to \(F_4\)
The author investigates minimal parabolic systems with diagram \(\circ\diagrbar\circ\overset\sim\diagrBar\circ\diagrbar\circ\). This means she studies groups \(G=\langle P_1,P_2,P_3,P_4\rangle\), \(B:=P_1\cap P_2\cap P_3\cap P_4\), \(B_G=1\), \(P_i/B_{P_i}\cong\Sigma_3\) and for \(G_i=\langle P_j\mid j\neq i\rangle\) we have \(G_1/B_{G_1}\cong G_4/BG_4\cong 3^7\text{Sp}(6,2)\) and \(G_2/B_{G_2}\cong G_3/B_{G_3}\cong\Sigma_3\times L_3(2)\). In a first step she establishes generators and relations for the group \(G\). These look very similar to those of the finite simple group \(F_4(2)\). Hence the main result of the paper is that there is a homomorphism \(\varphi\) from \(G\) onto \(F_4(2)\) mapping the \(G_i\) onto the corresponding maximal parabolics of \(F_4(2)\). In [Beitr. Algebra Geom. 38, No. 2, 337-342 (1997; Zbl 0890.51010)] the author constructed an example \(G\) as a nonsplit extension of an elementary abelian group of order \(3^{883}\) by \(F_4(2)\). In [J. Algebra 195, No. 2, 357-366 (1997; Zbl 0918.51017)] she showed that if the kernel of \(\varphi\) above is abelian it is of size \(3^{883}\). Hence the two main open questions are: (1) Is \(G\) finite? (2) If \(G\) is finite, is the kernel of \(\varphi\) abelian?
scientific article; zbMATH DE number 1247567