On Hadamard groups with relatively large $2$-subgroup

Abstract

A Hadamard group is any group of order $4u^2$ that contain a difference set. In this paper we obtain some new conditions for Hadamard groups with relatively large $2$-subgroup. We use norm invariant polynomials $f(\varepsilon) \in \mathbb{Z}[\varepsilon], \ |f(\varepsilon^t)|=const.$, where $\varepsilon$ is root of unity of order $2^n.$ Necessary condition on a size of normal cyclic $2$-subgroup are given. Also, we have covered cases when $2$-subgroup has generators similar to a modular or dihedral $2$-group. Additionally, we construct such two infinite series of groups. Obtained results are natural generalization of a case when entire group is $2$-group.