2-(v, k, 1) designs admitting a primitive rank 3 automorphism group G = T:G_0, where G_0 belongs to the Extraspecial Class, or to the Exceptional Class of Liebeck'’s Theorem (1987) are classifi…ed.

2-(v,k,1) Designs with a point-primitive rank 3 automorphism group of affine type are investigated and several new examples are provided

The affine planes admitting a collineation group with a transitive action on the line at infinity are investigated and an essentially complete classification is achieved when the group is almost simple. A rather deep result is also obtained when the action on the line at infinity is faithful and primitive.

For each q=2^m, m≥1, an infinite class of 2-(q^4+q+1,1) designs admitting PSU(3,q) as an automorphism group is constructed. Such designs are shown to contain embedded Desarguesian projective planes of order q and a PSU(3,q)-invariant oval.

In this paper it is shown that any unital embedded either in PG(2,q^2) or in the Hughes plane of order q^2, which is invariant under PGL(2,q)×σ, where σ is the Frobenius involution, is Hermitian or classical Rosati, respectively.

The structure of the fix bundle free automorphism groups of inversive planes of odd order is determined. As a special case of our main result, the automorphism groups with a transitive action on the points of an inversive plane of odd order are essentially determined, and the plane is shown to be miquelian when these have no non-trivial normal subgroups of odd order.

The general structure of collineation groups with a faithful transitive action on the line at infinity of a finite affine plane is determined. Moreover, the projective planes admitting a collineation group which fixes a point and acts transitively on a blocking set are classified when the group is of even order.