A note on two-generator groups (Q1263670)

The author studies the property (*): For any \(g\in G\) there is some \(h\in G\) such that \(<g,h>=G\) and either g or h is an involution. He shows that the Suzuki groups Sz(q) have property (*). Furthermore he shows that a simple Chevalley group over a field of odd characteristic having property (*) has to be isomorphic to PSL(2,q), for some odd q. This is done by using the known structure of the subgroups of Chevalley groups generated by two root elements, which usually are not the whole group.