A construction of difference sets in high exponent 2-groups using representation theory

Davis, James A. ; Smith, Ken

Journal of Algebraic Combinatorics, Tome 3 (1994) no. 2, pp. 137-151.

Résumé

Summary: 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 ($2 ^{2 d+2}, 2 ^{2 d+1}\pm 2 ^{ d }, 2 ^{2 d }\pm 2 ^{ d }$). 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 $2 ^{2 d+2}$ 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.

Détail

Keywords: difference set, representation theory, abelian group, nonabelian group