On split \((B,N)\)-pairs of rank 2
Let \(G^*\) be a finite group of Lie type of characteristic \(p\) and Lie rank \(m\). Then the associated Tits system \((G^*,B^*,N^*,S^*)\) has rank \(m\) and is saturated \((H^* = \bigcap_{N^*}B^{*n})\) and \(B^* = H^*O_ p(B^*)\). Suppose also that \(m = 2\) and let \(P^*_ 1\), \(P^*_ 2\) be the maximal parabolic subgroups of \((G^*,B^*,N^*,S^*)\). Also let \(U^*_ i = O_ p(P^*_ i)\) and let \(P^*_ i = K^*_ i U^*_ i\) be a Levi decomposition of \(P^*_ i\) for \(i = 1,2.\) Suppose that \(G^*\), as above, has no subgroups of index 2. Suppose that the finite group \(G\) also has a Tits system \((G,B,N,S)\) of rank 2 that is saturated and such that \(B = HO_ p(B)\). Let \(P_ 1\), \(P_ 2\) be the maximal parabolic subgroups of \((G,B,N,S)\). This paper is devoted to proving that \(G\) is isomorphic to \(G^*\) if \(P_ 1\), \(P_ 2\) satisfy a set of conditions satisfied by \(P^*_ 1\), \(P^*_ 2\) (so that \(P_ 1\), \(P_ 2\) ``resemble'' \(P^*_ 1\), \(P^*_ 2\)). Among these conditions are: (a) \(| B| = | P_ 1 \cap P_ 2| = | B^*|\), (b) \(H = B\cap N\) is an abelian \(p'\)-group and (c) \(P_ i = O_ p(P_ i)K_ i\) where \(H \leq K_ i\) and, for suitable indexing, \(K_ i\) is isomorphic to a Levi factor of \(P^*_ i\) for \(i = 1,2\). These results are used in the authors' work on revising the classification of finite simple groups.
scientific article; zbMATH DE number 202992