Construction of Relative Difference Sets in p-groups

Authors

James A. Davis, University of RichmondFollow

DOI

10.1016/0012-365X(92)90034-D

Abstract

Jungnickel (1982) and Elliot and Butson (1966) have shown that (p^j+1,p,p^j+1,p^j) relative difference sets exist in the elementary abelian p-group case (p an odd prime) and many 2-groups for the case p = 2. This paper provides two new constructions of relative difference sets with these parameters; the first handles any p-group (including non-abelian) with a special subgroup if j is odd, and any 2-group with that subgroup if j is even. The second construction shows that if j is odd, every abelian group of order p^j+2 and exponent less than or equal to p^(j+3)/2 has a relative difference set. If j is even, we show that every abelian group of order 2^j+2 and exponent less than or equal to 2^(j+4)/2 has a relative difference set except the elementary abelian group. Finally, Jungnickel (1982) found (p^i+j,pⁱ,p^i+j,p^j) relative difference sets for all i, j in elementary abelian groups when pis an odd prime and in ⁱ₄×^j₂ when p = 2. This paper also provides a construction for i+j even and i⩽j in many group with a special subgroup. This is a generalization of the construction found in a submitted paper.

Document Type

Article

Publication Date

5-25-1992

Publisher Statement

Copyright © 1992, Elsevier. This article first appeared in Discrete Mathematics: 103:1 (1992), 7-15.

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Recommended Citation

Davis, James A. "Construction of Relative Difference Sets in P-groups." Discrete Mathematics 103, no. 1 (May 25, 1992): 7-15. doi: 10.1016/0012-365X(92)90034-D.