On automorphism groups of divisible designs acting regularly on the set of point classes

Let ( P , B ) be an ( m , u , k , λ ) -divisible design and let G be a subgroup of Aut ( P , B ) . We say G is an SCT ( m , u , k , λ ) automorphism group of ( P , B ) if G is semiregular on P ∪ B and regular on the set of point classes of ( P , B ) . In this paper we show that each SCT ( m , u , k , λ ) automorphism group corresponds to a certain special kind of matrix, which we call an SCT ( m , u , k , λ ) matrix over G . Using such matrices we construct ( m , u , k , λ ) -divisible designs. We also consider the close connection between SCT automorphism groups and relative difference sets. As an application we generalize the notion of planar functions and show that an arbitrary p -group may be the forbidden subgroup of a semiregular relative difference set.