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 + 2 has 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 + 2 with exponent 23t + 2. Thus a nonabelian 2-group G with a Hadamard difference set can have exponent |G|3/4 asymptotically. Previously the highest known exponent of a nonabelian 2-group with a Hadamard difference set was |G|1/2 asymptotically. We use representation theory to prove that the group has a difference set.
Copyright © 1998, Academic Press. This article first appeared in Journal of Algebra: 199:1 (1998), 62-87.
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Davis, James A., and Joel E. Iiams. "Hadamard Difference Sets in Nonabelian 2-Groups with High Exponent." Journal of Algebra 199, no. 1 (January 1, 1998): 62-87. doi:10.1006/jabr.1997.7197.