The rigidity method and applications /

The Inverse Problem of Galois Theory is discussed. In a specific form, the problem asks whether every finite group occurs as a Galois group over Q . An intrinsically group theoretic property called rigidity is described which confirms that many simple groups are Galois groups over Q . Connections between rigidity and geometry are described and applications of rigidity are provided. In particular, after describing some of the theory of groups of Lie type, the rigidity criterion is applied to the exceptional Lie type groups G2(p), for primes p > 5. With the confirmation of a rationality condition, this establishes that G2(p) occurs as a Galois group over Q for all p > 5. Furthermore, the conjugacy classes which arise in the proof of rigidity for G2( p) are explored in detail, in the hope that a new proof might be produced which would illuminate the geometry associated to this rigid situation