Transitive permutation groups of finite order areviewed as linear groups over fields of characteristicp > 0 by having the group permute the basis elemerits of avector space M. The decomposition of M into the direct sumof invariant subspaces is investigated, and criteria givenfor whether M is decomposable, and if it is, how manydirect summands occur, in the special case the group hasrank 3, i.e., it has 3 orbits on ordered pairs of points.In the case that each orbit is self-paired, M decomposesinto the maximum possible number of indecomposables, andthe group has every p'-element conjugate to its inverse,irreducibility results are obtained for theindecomposables. This last result holds for any rank. Itapplies in particular to the symmetric and thence to thealternating groups, which enables us to describe certainmodular irreducibles of these groups.
