Rational rigidity and the sporadic groups

Abstract

The question of which finite groups can or do occur as the Galois group of a Galois extension of the rational numbers is one which has been studied since early in the twentieth century. It is conjectured that all finite groups do occur as Galois groups over the rationals: this is known to be true for all solvable groups. (See Jacobson [73 for brief historical notes.) In a breakthrough paper Thompson [9] has defined the new concept of rigidity of a finite group. He then proves that if a group G, with trivial centre, is rigid then an associated group G* is a Galois group over the rationals. In this paper it is shown that 14 of the 26 sporadic simple groups are Galois groups over the rationals. The method used involves the complex character table and knowledge of maximal subgroups of the group. In some cases where the maximal subgroups are not known the classification of finite simple groups is used. Section 1 contains the results, Section 2 describes the notation used and Section 3 details the proof for each of the 14 groups separately.