On primitivity and reduction for flag-transitive symmetric designs

We present some results on flag-transitive symmetric designs. First we see what conditions are necessary for a symmetric design to admit an imprimitive, flag-transitive automorphism group. Then we move on to study the possibilities for a primitive, flag-transitive automorphism group, and prove that for λ⩽3, the group must be affine or almost simple, and finally we analyse the case in which a biplane admits a primitive, flag-transitive automorphism group of affine type.