#### DOI

10.1023/A:102244682256

#### Abstract

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^{2d+2}, 2^{2d+1} ±2^{d}, 2^{2d}±2^{d}). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2^{d+2}. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^{2d+2} with exponent 2^{d+3} . We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.

#### Document Type

Article

#### Publication Date

4-1994

#### Publisher Statement

Copyright © 1994, Kluwer Academic Publishers. This article first appeared in *Journal of Algebraic Combinatorics*: 3:2 (1994), 137-151.

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#### Recommended Citation

Davis, James A., and Ken Smith. "A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory." *Journal of Algebraic Combinatorics* 3, no. 2 (April 1994): 137-51. doi: 10.1023/A:1022446822561.