Generating finite groups with maximal subgroups of maximal subgroups (Q1903021)

The author develops a theory of \(\gamma\)-triples \((G,M, H)\) where \(G\) is a finite group with proper subgroups \(H< M< G\) such that \(\langle H,g \rangle \cap M= H\) for all \(g\in G\setminus M\). He proves that in this situation the \(M\)-core \(H_M\) of \(H\) is subnormal in \(G\) and \(H/H_M\) is cyclic. If in addition \(H_G= 1\) and \(H_M< H\), then there is a prime \(p\) such that \(O_p (G) H=G\), \(O_p (G)\cap H= H_M\) (and \(G\) is soluble), also \(H\cap H^x\leq H_M\) for all \(x\in G\setminus H\) as in Wielandt's theorem; however \(G\) need not be a Frobenius group. These results can be applied to show that in a simple group \(G\) with a maximal subgroup \(H\) of a maximal subgroup \(M\) of \(G\) there is an element \(g\) with \(\langle H,g \rangle =G\). Fresh and detailed group theoretical arguments make for pleasant reading.