Application of Gaschütz' Theorem to relative difference sets in non-abelian groups

2003 ◽  
Vol 11 (5) ◽  
pp. 307-311 ◽  
Author(s):  
John C. Galati
Keyword(s):  
Abelian Groups ◽  
Difference Sets ◽  
Relative Difference ◽  
Relative Difference Sets

scholarly journals Enumeration of (16,4,16,4) Relative Difference Sets

10.37236/2776 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
David Clark ◽  
Vladimir D. Tonchev
Keyword(s):  
Normal Subgroup ◽  
Abelian Groups ◽  
Difference Sets ◽  
Relative Difference ◽  
Complete Enumeration ◽  
Relative Difference Sets ◽  
Nonabelian Groups

A complete enumeration of relative difference sets (RDS) with parameters $(16,4,16,4)$ in a group of order 64 with a normal subgroup $N$ of order 4 is given. If $N=Z_4$, three of the 11 abelian groups of order 64, and 23 of the 256 nonabelian groups of order 64 contain  $(16,4,16,4)$ RDSs. If $N=Z_2 \times Z_2$, nine of the abelian groups and 194 of the non-abelian groups of order 64 contain  $(16,4,16,4)$  RDSs.

Divisible Semiplanes, Arcs, and Relative Difference Sets

1987 ◽  
Vol 39 (4) ◽  
pp. 1001-1024 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Projective Plane ◽  
Difference Sets ◽  
Difference Set ◽  
Relative Difference ◽  
List Type ◽  
Relative Difference Set ◽  
Large Collection ◽  
Baer Subplane ◽  
Relative Difference Sets ◽  
Definition Of

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:(i) S is the empty set.(ii) S consists of a line L with all its points and a point p with all the lines through it.(iii) S is a Baer subplane of Π.We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

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