Kantentransitive Graphen und Gruppen vom Rang 2. (Edge transitive graphs and groups of rank 2) (Q1075427)

This paper is a contribution to the investigations on ''pushing up'' results, which started with \textit{D. M. Goldschmidt}'s paper on trivalent graphs [Ann. Math., II. Ser. 111, 377-406 (1980; Zbl 0475.05043)]. In the paper under review groups G are investigated, which satisfy the following properties: (1) \(G=<M_ 1,M_ 2>\), \(| M_ i| <\infty.\) (2) If \(N\trianglelefteq G\), \(N\leq M_ 1\cap M_ 2\), then \(N=1.\) (3) \(| M_ i/(M_ 1\cap M_ 2)| =q_ i+1\), \(q_ i\) a 2- power. (4) There are normal subgroups \(N_ i\trianglelefteq M_ i\), such that for \(R_ i=\cap_{x\in M_ i}(M_ i\cap M^ x_ j)\), \(\{i,j\}=\{1,2\}\), we have \(N_ 1/R_ 1\simeq U_ 3(^ 3\sqrt{q_ 1})\), \(q_ 1>8\) and \(N_ 2/R_ 2\simeq U_ 3(^ 3\sqrt{q_ 2})\), \(Sz(^ 2\sqrt{q_ 2})\), or \(L_ 2(q_ 2)\) \(q_ 2>8\) respectively \(q_ 2\geq 4.\) Any of the groups \(X\simeq U_ 3(q)\times A\), \(U_ 3(q)\wr A\), \(A\wr U_ 3(q)\), or \(U_ 5(q)\), \(A\in \{U_ 3(r)\), Sz(r), \(L_ 2(r)\}\); q,r 2-powers contains precisely two classes of maximal 2-local subgroups with representatives \(X_ 1\), \(X_ 2\) such that \(X_ i\) contains a Sylow 2- subgroup of X. With the help of these \(X_ i's\) the author then defines for groups Y (X\(\leq Y\leq Aut(X))\) two classes of certain 2 local subgroups and calls them parabolic of type X (we omit the precise description). He then proves, that under hypotheses (1)-(4) \(M_ 1\) and \(M_ 2\) are parabolic of type X, where X is as above. For the methods and the significance of this paper the book of \textit{A. Delgado}, \textit{D. Goldschmidt}, and \textit{B. Stellmacher} [''Groups and graphs: new results and methods'' (1985; Zbl 0566.20013)] is a good reference.