On the Uniqueness Theorem (Q6600755)

One of the fundamental results of the Odd Order paper ([\textit{W. Feit} and \textit{J. G. Thompson}, Pac. J. Math. 13, 775-1029 (1963; Zbl 0124.26402)]) is that in a minimal simple group \(G\) of odd order an elementary Abelian subgroup \(R\) of \(G\) of type \((r,r)\) which is contained in an elementary abelian subgroup of type \((r,r,r)\) is contained in a unique maximal subgroup of \(G\). In a revised version of the Odd Order Paper, \textit{H. Bender} and \textit{G. Glauberman} ([Local analysis for the odd order theorem. Cambridge: Cambridge Univ. Press. (1994; Zbl 0832.20028)]) called this result the Uniqueness theorem.\N\NIn the paper under review the authors investigate whether the Uniqueness theorem holds in finite simple groups of Lie type of Lie rank one. They also introduce and examine the Strong Uniqueness theorem. They define the concept of containment minimality and prove that a non-abelian finite simple group is minimal if and only if it is containment minimal. Finally, they inquire when a minimal non-soluble group has a faithful doubly transitive representation.