Difference Sets Relative to Disjoint Subgroups

In their paper (1967, Math. Z.99, 53–75) P. Dembowski and F. C. Piper gave a classification of quasiregular collineation groups of finite projective planes. In the case (d) or (g) in their list the corresponding group, say G, has a subset D satisfying that (*) there exist mutually disjoint subgroupsH1, …, Hmof G such that the differencesd1 d−12 (d1≠d2∈D) contain each element outside ∪i Hiexactly λ times and no element of ∪i Hi. We note that if m=1, the notion is the same as relative difference sets introduced by J. E. H. Elliot and A. I. Butson (1966, Illinois J. Math.10, 517–531). In the case (d) or (g), (m, λ)=(2, 1) or (3, 1), respectively. In this article we study groups with the property (*). Under some additional condition we give a result on their group theoretic structure (Theorem 4.1). Moreover, we study the case that {H1, …, Hm} is a partial spread of G (Theorem 4.7).