On difference sets in high exponent 2-groups

• Published in: Journal of algebraic combinatorics 2013-12, Vol.38 (4), p.785-795
• Main Authors: Mandić, Joško, Pavčević, Mario Osvin, Tabak, Kristijan
• Format: Article
• Language: English
• Subjects: Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-013-0425-1

We investigate the existence of difference sets in particular 2-groups. Being aware of the famous necessary conditions derived from Turyn’s and Ma’s theorems, we develop a new method to cover necessary conditions for the existence of (2 2 d +2 ,2 2 d +1 −2 d ,2 2 d −2 d ) difference sets, for some large classes of 2-groups. If a 2-group G possesses a normal cyclic subgroup 〈 x 〉 of order greater than 2 d +3+ p , where the outer elements act on the cyclic subgroup similarly as in the dihedral, semidihedral, quaternion or modular groups and 2 p describes the size of G ′∩〈 x 〉 or C G ( x )′∩〈 x 〉, then there is no difference set in such a group. Technically, we use a simple fact on how sums of 2 n -roots of unity can be annulated and use it to characterize properties of norm invariance (prescribed norm). This approach gives necessary conditions when a linear combination of 2 n -roots of unity remains unchanged under homomorphism actions in the sense of the norm.