Hadamard Difference Sets in Nonabelian 2-Groups with High Exponent

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 22 tq2 has a difference set if and only if the exponent of the group is less tq2 Ž than or equal to 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 24 tq2 with exponent 23 tq2. Thus a nonabelian 2-group G with a Hadamard difference set can have exponent < < 3r4 G asymptotically. Previously the highest known exponent of a nonabelian < <1r2 2-group with a Hadamard difference set was G asymptotically. We use representation theory to prove that the group has a difference set. Q 1998 Academic Press