Hadamard Difference Sets in Nonabelian 2-Groups with High Exponent Original Research Article

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 22t + 2has a difference set if and only if the exponent of the group is less than or equal to 2t + 2. In a previous work (R. A. Liebler and K. W. Smith,in“Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed 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 24t + 2with exponent 23t + 2. Thus a nonabelian 2-groupGwith a Hadamard difference set can have exponent G3/4asymptotically. Previously the highest known exponent of a nonabelian 2-group with a Hadamard difference set was G1/2asymptotically. We use representation theory to prove that the group has a difference set.