The natural character π of a finite transitive permutation group G has the form 1G + θ where θ is a character which affords a rational representation of G. We call G a QI-group if this representation is irreducible over Q. Every 2-transitive group is a QI-group, but the latter class of groups is larger. It is shown that every QI-group is 3/2transitive and primitive, and that it is either almost simple or of affine type. QI-groups of affine type are completely determined relative to the 2… CONTINUE READING