Fully invariant transformations and associated groups

It is well known that the symmetric group S-n together with one idempotent of rank n - 1 on a finite n-element set N serves as a set of generators for the semigroup T-n of all the total transformations on N. It is also well k nown that the singular part Sing(n) of T-n can be generated by a set of ide mpotents of rank n - 1. The purpose of this paper is to begin an investigat ion of the way in which Sing, and its subsemigroups can be generated by the conjugates of a subset of elements of T-n by a subgroup of S-n. We look fo r the smallest subset of elements of T-n that will serve and, corresponding ly, for a characterization of those subgroups of S-n that will serve. Using some techniques from graph theory we prove our main result: the conjugates of a single transformation of rank n - 1 under G suffice to generate Sing( n), if and only if G is what we define to be a 2-block transitive subgroup of S-n.