Let G be a group, p a fixed prime, and B a finite p-closed subgroup of G. A set \(\Omega =\{P_ 1,...,P_ n\}\) is called a parabolic system of rank n for G if the following holds for \(1\leq i\leq n:\) (1) \(B=N_{P_ i}(O_ p(B))\), (2) B is contained in a unique subgroup of \(P_ i\), (3) \(O_ p(P_ i)\neq 1\), (4) \(G=<\Omega >\) but \(G\neq <\Omega '>\) for \(\Omega\) ’\(\subsetneqq \Omega.\)
In the case of a finite Chevalley group G of characteristic p \(\Omega\) is the set of minimal parabolic subgroups containing a Borel subgroup B. The examples of parabolic systems for \(p=2\) in sporadic finite simple groups [see M. Ronan and G. Stroth, Eur. J. Comb. (to appear)] suggest to study parabolic systems satisfying
(A) \(\Omega =\{M_ 1,M_ 2\}\), (i) \(M_ 1/O_ 2(M_ 1)\simeq \Sigma_ 5\), \(M_ 2/O_ 2(M_ 2)\simeq \Sigma_ 3\), (ii) If \(N\subseteq M_ 1\cap M_ 2\), \(N\trianglelefteq G\), then \(N=1\), (iii) \(C_{M_ i}(O_ 2(M_ i))\subseteq O_ 2(M_ i)\), \(i=1,2.\)
A parabolic system \(\{M_ 1,M_ 2\}\) is of type X if the finite group X contains a parabolic system \(\{\) \(\tilde M_ 1,\tilde M_ 2\}\) with \(\tilde M_ i\simeq M_ i\). Under hypothesis (A) it is shown, that the parabolic system is of type \(Aut(U_ 4(2))\), \(M_{22}\), \(Aut(M_{22})\), HS, Aut(HS), Ly, or Ru. The proof of this result depends on a graph theoretic approach originally introduced by D. Goldschmidt. More material related to these concepts can be found in [A. Delgado, D. Goldschmidt, B. Stellmacher, Groups and graphs (1985; Zbl 0566.20013)].