Difference Sets Relative to Disjoint Subgroups

✍ Scribed by Yutaka Hiramine

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 133 KB
• Volume: 88
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1999.2990

✦ Synopsis

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 (V) there exist mutually disjoint subgroups H 1 , ..., H m of G such that the differences d 1 d &1 2 (d 1 {d 2 # D) contain each element outside i H i exactly * times and no element of i H i . 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, *)=(2, 1) or (3, 1), respectively. In this article we study groups with the property (V). Under some additional condition we give a result on their group theoretic structure (Theorem 4.1). Moreover, we study the case that [H 1 , ..., H m ] is a partial spread of G (Theorem 4.7).

1999 Academic Press

Here

Remark 1.1. Let G be a group and let D be any subset of G. Then D D &1 @ =* 0 S 0 @+* 1 S 1 @+ } } } +* n S n @ for a suitable partition G=S 0 _ S 1 _ } } } _ S n (S 0 =[1]) of G and non-negative integers * 0 , * 1 , ..., * n . Therefore,