#### 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},*p*^{i},*p*^{i+j},*p*^{j}) relative difference sets for all *i*, *j* in elementary abelian groups when *p*is an odd prime and in ^{i}_{4}×^{j}_{2} 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.