Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then < G,a > \ G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a(g) = g(-1)ag of a by elements of G generate a semigroup denoted by (a(g) vertical bar g is an element of G). We classify the finite permutation groups G on a finite set X such that the semigroups < G, a >, < G, a > backslash G, and < a(g) vertical bar go G) are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups (G, a) backslash G and (a(g) vertical bar g is an element of G) are generated by their idempotents for all non-invertible transformations of X. (C) 2011 Elsevier Inc. All rights reserved.