On difference sets in high exponent 2-groups

Mandić, Joško ; Pavčević, Mario Osvin ; Tabak, Kristijan

Journal of Algebraic Combinatorics, Tome 38 (2013) no. 4, pp. 785-795.

Résumé

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 ($2^{2d+2},2^{2d+1} - 2^{d },2^{2d } - 2^{d }$) difference sets, for some large classes of 2-groups. If a 2-group $G$ possesses a normal cyclic subgroup $\langle x\rangle $ 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^{\prime}\cap \langle x\rangle $ or $C _{G }(x)^{\prime}\cap \langle x\rangle $, 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.

Détail

Classification : 05B10, 20C20
Keywords: difference set, norm invariance, Hadamard group, modular 2-group, group representation